vulnerability to sleep deprivation: a drift diffusion model perspective

Vulnerability to Sleep Deprivation: ADrift Diffusion Model Perspective

by

Amiya Patanaik

Thesis submitted in partial fulfilment of the requirements for theaward of

Doctor of Philosophy

in theSchool of Computer Engineering

April 28, 2015

Dedicated to my mother.

“The ultimate arbiter of truth is experiment, not the comfort one derives fromone’s a priori beliefs, nor the beauty or elegance one ascribes to one’s theoreticalmodels.”

Lawrence M. Krauss

Acknowledgments

It is a great pleasure for me to thank the people who have helped me during

my PhD. First and foremost, I express my sincere gratitude and indebtedness to

my former supervisor Dr. Vitali Zagorodnov under whose esteemed guidance and

supervision, most of the work has been done. Without his consistent support and

help, the research would not have been so enriching and fulfilling. I would also like

to express my heartfelt gratitude to my supervisor Dr. Kwoh Chee Keong, who

gracefully took me under his supervision at a critical juncture. His feedback and

support helped me immensely in giving shape to my research. I am also thankful

to him for always keeping his door open for me to discuss my research problems.

His constant support and motivation kept me going.

Secondly, I would like to thank my collaborators Dr Michael Chee and Dr.

Joshua Gooley heading the Cognitive Neuroscience Laboratory and Chronobiology

and Sleep Laboratory respectively at DUKE-NUS Graduate Medical School as

well as other members of their labs who took their valuable time to guide and

help me throughout the whole period. I am indebted to them for providing data

collected from long and arduous experiments and providing invaluable insights

from a clinical perspective.

Last but not the least; I would like to thank my parents, A. R. Patanaik and

Suniti Patanaik, my brother Subhan Patanaik and my fiancee Litali Mohapatra

for all the help, motivation, love and support that they have provided. It would

have been impossible to complete this thesis without the confidence that I had

their support, and help, during the entire period of my PhD. Finally, I would like

to thank my friends, and colleagues, especially Vishram Mishra, Gyanendu Sahoo,

Sahil Bansal, Ankit Das, Sateesh Babu, Akhil Garg, Nitin Sharma and Gigi Chi

Ting Au-Yeung for adding life to the graduation years. I thoroughly enjoyed the

time I spent with you all, having various technical and non-technical discussions.

vii

Contents

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii

List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii

List of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Wearable devices, Smartphones and Sleep . . . . . . . . . 3

1.2 Research statement, scope and objectives . . . . . . . . . . . . . . 4

1.3 Issues and Challenges . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Review of Literature 7

2.1 Sleep Deprivation and Vulnerability . . . . . . . . . . . . . . . . . 7

2.1.1 Quantifying Performance: The PVT . . . . . . . . . . . . 7

2.1.2 Performance Degradation with SD . . . . . . . . . . . . . 8

2.1.3 Vulnerability to Sleep Deprivation . . . . . . . . . . . . . . 10

2.1.4 Model of Sleep Regulation . . . . . . . . . . . . . . . . . . 11

2.2 Mental Chronometry . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 Standard RT Analysis . . . . . . . . . . . . . . . . . . . . 14

2.2.2 Statistical Model of RT distribution . . . . . . . . . . . . . 14

2.3 Models of Perceptual Decision Making . . . . . . . . . . . . . . . 16

2.3.1 Signal Detection Theory . . . . . . . . . . . . . . . . . . . 16

2.3.2 Accumulator Models . . . . . . . . . . . . . . . . . . . . . 16

2.3.3 Perceptual Decision Making: Neural Basis . . . . . . . . . 17

2.4 The Drift Diffusion Model . . . . . . . . . . . . . . . . . . . . . . 18

ix

2.4.1 Diffusion Process Applied to Perceptual Decision Making . 21

2.4.2 Extending DDM to The PVT . . . . . . . . . . . . . . . . 23

2.4.3 Experiments Supporting The DDM . . . . . . . . . . . . . 24

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 The One Choice DDM: Simulation and Estimation 27

3.1 Approximate the Diffusion Process . . . . . . . . . . . . . . . . . 27

3.2 Simulating the DDM . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.1 A simplified version of DDM . . . . . . . . . . . . . . . . . 29

3.2.2 A mixed simulation method . . . . . . . . . . . . . . . . . 31

3.2.3 Simulating PVT . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3 DDM parameter estimation . . . . . . . . . . . . . . . . . . . . . 32

3.3.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . 32

3.3.2 DDM: identifiability issues . . . . . . . . . . . . . . . . . . 33

3.3.3 Current estimation method . . . . . . . . . . . . . . . . . 33

3.3.4 MCMC-MLE . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4 Improvements in Estimation . . . . . . . . . . . . . . . . . . . . . 39

3.4.1 DDM: Closed form solution . . . . . . . . . . . . . . . . . 39

3.5 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.5.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.5.2 Comparison of simulation methods . . . . . . . . . . . . . 43

3.5.3 Comparison of estimation methods . . . . . . . . . . . . . 43

3.6 Cramer-Rao Lower Bound . . . . . . . . . . . . . . . . . . . . . . 44

3.6.1 CRLB estimates using simulations . . . . . . . . . . . . . . 46

3.6.2 Error estimates: Results . . . . . . . . . . . . . . . . . . . 46

3.7 DDM: Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4 DDM Parameters and Differential Vulnerability to Sleep Deprivation 50

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2.1 Subjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2.2 Experimental Details . . . . . . . . . . . . . . . . . . . . . 52

4.2.3 DDM parameter Estimation . . . . . . . . . . . . . . . . . 52

4.2.4 Statistical analyses . . . . . . . . . . . . . . . . . . . . . . 52

4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3.1 Identification of vulnerable and resistant subjects . . . . . 53

4.3.2 Effects of state and group on diffusion parameters . . . . . 54

4.3.3 Predicting vulnerability from baseline data . . . . . . . . . 55

4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.4.1 Effect of sleep deprivation on diffusion parameters . . . . . 56

4.4.2 Difference in diffusion parameters between vulnerable and

resistant subjects . . . . . . . . . . . . . . . . . . . . . . . 57

4.4.3 Interaction effect of state and group on diffusion parameters 59

4.4.4 Possible neurocognitive accompaniments of reduced diffusion

drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.4.5 Predictive value of diffusion model parameters . . . . . . . 60

4.4.6 Strengths and limitations . . . . . . . . . . . . . . . . . . . 60

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5 Classifying Vulnerability to Sleep Deprivation Using Baseline

Measures of Psychomotor Vigilance 63

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.2.1 Subjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.2.2 Sleep deprivation procedures . . . . . . . . . . . . . . . . . 65

5.2.3 Assessment of vulnerability to sleep deprivation . . . . . . 65

5.2.4 RT Derived Features . . . . . . . . . . . . . . . . . . . . . 67

5.2.5 Feature Selection . . . . . . . . . . . . . . . . . . . . . . . 69

5.2.6 Classifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.2.7 Statistical Analyses . . . . . . . . . . . . . . . . . . . . . . 72

5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.3.1 Model performance in Dataset 1 . . . . . . . . . . . . . . . 73

5.3.2 Model performance following training on Dataset 1 and testing

on Dataset 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.3.3 Model performance in Dataset 2 . . . . . . . . . . . . . . . 74

5.3.4 Reproducibility of classification across testing episodes in

the same participants . . . . . . . . . . . . . . . . . . . . . 74

5.3.5 Classification using the most sensitive PVT measures . . . 74

5.3.6 DDM variability across visits and with time; computational

performance . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.4.1 Features most useful for discriminating vulnerable and resistant

participants . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.4.2 Classification reliability and reproducibility . . . . . . . . . 82

5.4.3 Differences between datasets . . . . . . . . . . . . . . . . . 83

5.4.4 Definition of vulnerability to total sleep deprivation . . . . 83

5.4.5 Further improvements in classification . . . . . . . . . . . 84

5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6 Baseline resting state connectivity differences between individuals

vulnerable and resistant to sleep deprivation 85

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.2.1 BOLD - fMRI . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.2.2 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . 86

6.2.3 Data Pre-Processing . . . . . . . . . . . . . . . . . . . . . 87

6.2.4 General Linear Model-Finding regions of significant task

activations . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.2.5 Statistical ThresholdingProblem of Multiple Comparisons . 88

6.2.6 Functional Connectivity and Brain Networks . . . . . . . . 90

6.2.7 Activations in the absence of task - resting state networks 91

6.2.8 Finding Group and Subject RSNs . . . . . . . . . . . . . . 93

6.3 Materials and methods . . . . . . . . . . . . . . . . . . . . . . . . 95

6.3.1 Participants . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.3.2 Study procedure . . . . . . . . . . . . . . . . . . . . . . . 96

6.3.3 Imaging methods . . . . . . . . . . . . . . . . . . . . . . . 96

6.3.4 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.4.1 Functional connectivity differences between vulnerable and

resistant groups in RW . . . . . . . . . . . . . . . . . . . . 98

6.4.2 Functional connectivity differences between vulnerable and

resistant groups in SD . . . . . . . . . . . . . . . . . . . . 98

6.4.3 Functional connectivity changes from RW to SD in vulnerable

and resistant groups . . . . . . . . . . . . . . . . . . . . . 99

6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.5.1 Reduced functional connectivity in the ventral right PPC

region as a marker of vulnerability to SD . . . . . . . . . . 102

6.5.2 Role of frontoparietal network . . . . . . . . . . . . . . . . 102

6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7 Summary and Outlook 104

7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7.2.1 Absolute scale for vulnerability . . . . . . . . . . . . . . . 106

7.2.2 Prediction under field conditions . . . . . . . . . . . . . . 106

7.2.3 Improving classification accuracy . . . . . . . . . . . . . . 107

7.2.4 Understanding the mechanics of differential vulnerability . 108

7.3 Conclusion and Final Thoughts . . . . . . . . . . . . . . . . . . . 108

A Effective PVT Data Acquisition 110

A.1 Existing PVT Data Acquisition Systems . . . . . . . . . . . . . . 110

A.2 Technical Challenges . . . . . . . . . . . . . . . . . . . . . . . . . 111

A.3 Design Considerations and Possible solutions . . . . . . . . . . . . 112

A.4 Quick PVT: A Cross Platform PVT Software Solution . . . . . . 113

A.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Publication 115

References 116

List of Figures

1.1 A: Most popular wearable biosensor devices as of 2014. Nike, Fitbit

and Jawbone combined constitute 97% market share. B: Number

of smartphone users across time. The dotted lines are projections. 3

2.1 A subject taking the PVT in a reclined position. Inset: A portable

PVT monitor, courtesy of artisan-scientific.com. . . . . . . . . . . 9

2.2 PVT RT in milli-seconds for a representative subject. Even after

12 and 84 hours of SD, the subject was able to perform at baseline

levels albeit, with a clear increase in the variability in performance.

Courtesy of Doran et al. (2001). . . . . . . . . . . . . . . . . . . . 10

2.3 Subjects were administered the Karolinska Sleepiness Scale (KSS-

1), the Word Detection Task (WDT), and the Psychomotor Vigilance

Task (PVT) after they underwent 24 hours of sleep deprivation

(SD) on two separate occasions. The KSS is a subjective measure

of vigilance, the WDT is a test of cognitive processing ability and

the PVT is an objective measure of vigilance. The 21 subjects

were arbitrarily labeled from A through U. Boxes demarcate first

exposure to SD and diamonds demarcate the second. While the

ranking between subjects across repeated trials for all three tasks

remain stable, there is no direct correspondence between ordering

in one task with ordering in another. Courtesy of H. Van Dongen,

Baynard, et al. (2004). . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Two-process model of sleep regulation. Process S indicates the

homeostatic built-up of sleep pressure and Process C represents

the circadian rhythm. The difference between the two processes

quantify the sleep pressure. . . . . . . . . . . . . . . . . . . . . . 13

2.5 A sample reaction time (RT) distribution. RT distributions tend

to be strongly skewed towards right. . . . . . . . . . . . . . . . . 15

xiv

2.6 General framework of sequential analysis model. . . . . . . . . . . 17

2.7 Top: The accumulator model assumes that the perceptual decision

making is a serial process progressing from perception to action.

Bottom: Diffusion model is an accumulator model for the neurobiology

of decision making inspired by experiments. The figure shows the

model for a two choice visual task (house or face?). It assumes

that decisions are formed by continuously accumulating sensory

information until one of the response criteria are met (a or -b).

The rate of information accumulation µ, models the firing rate of

the neurons involved in the decision process. The rate is lower

(red path) for a difficult task with low sensory evidence and higher

(green path) for a easy task (high sensory information). The moment

to moment fluctuations in the sample path reflect the noise in the

decision process. Figure adapted from H. Heekeren et al. (2004). . 19

2.8 A diffusion process with constant mean β and variance α is equivalent

to a Wiener process. Shown here are sample paths for such a

diffusion process modeling task with one and two hypothesis respectively. 20

2.9 Components of reaction time. . . . . . . . . . . . . . . . . . . . . 21

2.10 Seven parameter diffusion model for two choice RT task. The non-

decision time and the starting point are assumed to be distributed

uniform. The drift varies from trial to trial according to a normal

distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.11 Five parameter diffusion model for one choice RT task. The non-

decision time is assumed to be distributed uniform. The drift varies

from trial to trial according to a normal distribution. . . . . . . . 24

2.12 Timeline shows the major findings in the context of our investigation.

While all findings are important, the ones marked in red are heavily

relied on in the current work. . . . . . . . . . . . . . . . . . . . . 26

3.1 Figure illustrates the standard simulation process for the DDM. . 29

3.2 Figure illustrates the DDM parameter estimation process as described

by Ratcliff & Van Dongen (2011). . . . . . . . . . . . . . . . . . . 34

3.3 Ratio of simulation speeds of random walk simulation to mixed

simulation method. The numbers shown are actual simulation speed

of the proposed mixed simulation method in seconds. As we move

from set 1 to 6, the mean drift goes on reducing. As a result the

proportion of sampled drift that is less than zero also increases.

This in-turn reduces the speed of mixed simulation. Under normal

circumstances, mixed simulation is two order of magnitudes faster

than random walk simulation. . . . . . . . . . . . . . . . . . . . . 43

3.4 Standard deviation of error as progressively more data is used for

estimation. The CRLB estimate shows the lowest possible error

achievable with the given amount of data. The analysis is done for

a fixed value of parameters that represent an average subject under

baseline condition. . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.1 Performance metrics measured on the evening before sleep deprivation

(ESD). Metrics include mean reaction time (RT), mean response

speed (RS=1/RT), total lapses and median RT for subjects vulnerable

(vul) and resistant (res) to sleep deprivation. Although the resistant

group performed better than the vulnerable group in the ESD state,

none of the RT metrics were statistically significantly different between

the two groups (smallest p=0.08 for mean RT). Error bars represent

one standard error of the mean. . . . . . . . . . . . . . . . . . . . 55

4.2 Mean estimated diffusion parameters. Estimated for vulnerable

(vul) and resistant (res) groups in the evening before sleep deprivation

(ESD) and after sleep deprivation (SD). In ESD, vulnerable subjects

showed significantly lower (p<0.05) mean diffusion drift compared

to resistant subjects. In addition, following SD, vulnerable subjects

also displayed significantly lower drift (p<0.001) and significantly

higher non-decision time (p<0.01). Both mean diffusion drift and

mean non-decision time showed statistically significant group by

state interaction (p<0.005 and p<0.05 respectively). Error bars

represent one standard error of the mean. . . . . . . . . . . . . . . 56

4.3 Receiver operating characteristic curve obtained by varying threshold

of a logistic regression based classifier. Diffusion parameters estimated

on the evening before sleep deprivation (ESD) state were used to

predict the vulnerability of the subjects to sleep deprivation. The

best operating point was found at a true positive (TP) rate of 65%

and false positive (FP) rate of 30.7%. The area under the curve

was 0.74. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.4 Expected median reaction time (RT) for different combinations

of mean normalized diffusion drift and mean non-decision time

generated by simulating 30 minutes of psychomotor vigilance task

(PVT). The variability parameters were fixed at the group average

values (η/a=2.62,st=52ms). Response times (as measured by median

RT) of two representative subjects, one vulnerable (vul) and the

other resistant (res) to SD were overlaid. Both subjects had the

same median RT (=256ms) in the baseline Evening Before Sleep

Deprivation (ESD) condition. . . . . . . . . . . . . . . . . . . . . 58

4.5 Mean reaction time in milliseconds across sessions. Error bars

represent one standard error of the mean. . . . . . . . . . . . . . . 61

5.1 (A) Dataset 1: Subjects (n=135) arrived at the laboratory at 7:30pm

and stayed awake overnight resulting in total sleep deprivation of

22 h. A 10-min PVT was administered every hour from 8:00pm

until 5:00am on the next morning. (B) Dataset 2: After an 8-h

opportunity for sleep, subjects underwent sleep deprivation in the

laboratory for at least 26 h. Every 2 h, subjects completed a 10-

min psychomotor vigilance task (PVT), indicated by the circles.

A subset of subjects (n=34) participated in a follow up session in

which two PVTs were taken in the mid-afternoon. In each dataset,

subjects were stratified into vulnerable and resistant groups by

performing a median split of PVT lapse data (reaction times >

500 ms) during the last session of sleep deprivation (red circles).

Two baseline PVT sessions (green circles) were used to build the

classifier for predicting vulnerability to sleep deprivation. . . . . . 66

5.2 Time course of PVT lapses in vulnerable and resistant groups for

(A) Dataset 1 and (B) Dataset 2. Inset: Individual traces show

the time course of lapses for each participant who underwent sleep

deprivation. Mean ± SEM are shown. . . . . . . . . . . . . . . . . 67

5.3 A four level decomposition of signal using discrete wavelet transform.

At each level, the resolution of the signal is halved. . . . . . . . . 70

5.4 Summary of the overall feature selection process. The process was

applied independently on both datasets. . . . . . . . . . . . . . . 72

5.5 Stratified 5-fold cross validation (CV5) accuracy of the classifier as

features ranked by minimal-redundancy-maximal-relevance (mRMR)

criterion were incrementally added to the feature set for (A) Dataset

1 and (B) Dataset 2. The top 8 features in each dataset (partitioned

by vertical line) were considered for the candidate feature set. The

feature set comprised of 5 DDM parameters (mean drift: ξ/a, across

trial variability in drift: η, mean non-decision time: Ter, variability

in non-decision time: St, drift signal to noise ratio: driftSNR);

9 standard reaction time (RT) metrics (mean RT, mean response

speed (RS), fastest 10% RT, median RT, slowest 10% RS, median

RT, standard deviation of RT: std RT, mean absolute deviation

of RT: MAD RT and the number of consecutive RTs that differ

by more than 250ms: ∆RT > 250); and 6 metrics derived from

spectral analysis of RTs (Mean Absolute Value of detail coefficient

at level 1 through level 6: MAV1 to MAV6). . . . . . . . . . . . . 76

5.6 Receiver operating characteristic (ROC) curves obtained by varying

the threshold of class membership probability of the SVM classifier

for (A) Dataset 1 and (B) Dataset 2 using the best set of features.

The best performing point on the ROC curve is marked with a gray

circle. Inset: confusion matrix, accuracy, sensitivity and specificity

at the best performing point. . . . . . . . . . . . . . . . . . . . . . 77

5.7 Performance of the classifier trained on Dataset 1 and tested on

Dataset 2 is demarcated by the gray circle on the receiver operating

characteristic (ROC) curve. The corresponding confusion matrix,

accuracy, sensitivity and specificity are presented below. A more

balanced classifier performance was obtained by changing the class

membership probability threshold from its default value (marked

with a black circle on the ROC). While the accuracy did not improve,

the sensitivity and specificity became more balanced. . . . . . . . 78

5.8 Estimated drift diffusion model (DDM) parameters for vulnerable

and resistant subjects estimated from baseline PVT sessions measured

across two study visits for Dataset 1. Of the 45 subjects who

participated in the first study, 34 returned for a follow up study

after at least 5 months following their initial visit to the laboratory.

Baseline individual differences in mean diffusion drift and variability

in non-decision time were reproducible across study visits. Vulnerable

subjects had lower mean drift (F1,32 = 11.6, P < 0.005) and

higher variability in non-decision time (F1,32 = 13.6, P < 0.001)

across both visits compared to resistant subjects. The across trial

variability in diffusion drift appeared to be the only DDM parameter

to be affected across the two studies (F1,32 = 4.65, P < 0.05).

The ‘#’ symbol denotes significant main effects of group on DDM

parameters. Mean ± SEM are shown. . . . . . . . . . . . . . . . . 79

5.9 Estimated drift diffusion model (DDM) parameters for vulnerable

and resistant subjects across the study period for Dataset 2. DDM

parameters were significantly affected by time of day. Vulnerable

subjects had lower mean diffusion drift (F1,43 = 4.03, P < 0.001)

and higher variability in non-decision time (F1,43 = 27.5, P <

0.001) irrespective of time elapsed since waking. Across trial variability

in drift and mean non-decision time showed interesting variations

over the period of the day. The combined effect of all the DDM

parameters was such that depending on the time of the day, the

parameters could act in opposite directions. The dotted vertical

line demarcates the usual bed time. Asterisk indicates statistically

significant difference (P < 0.05). . . . . . . . . . . . . . . . . . . . 80

6.1 fMRI data acquisition process. The spatial resolution of the scan

is h× w × d and temporal resolution is TR. . . . . . . . . . . . . 87

6.2 To find regions of significant task activation GLM is used to find out

the variability in measured signal explained by the task. Statistical

tests are done on the coefficients β to find voxels with significant

task activation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.3 Eight of the most common and consistent RSNs identified by ICA.

(A) RSN located in primary visual cortex; (B) extrastriate visual

cortex; (C) auditory and other sensory association cortices; (D)

the somatomotor cortex; (E) the default mode network (DMN),

deactivated during demanding cognitive tasks and involved in episodic

memory processes and self-referential mental representations; (F)

a network implicated in executive control and salience processing;

and (G,H) two right- and left-lateralised fronto-parietal RSNs spatially

similar to the bilateral dorsal attention network and implicated in

working memory and cognitive attentional processes. Courtesy of

Beckmann et al. (2005). . . . . . . . . . . . . . . . . . . . . . . . 92

6.4 spatial-Independent Component Analysis (sICA) decomposition of

group data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.5 In dual regression individual time-courses are estimated first, which

are then used to estimate individual spatial maps. Image courtesy

of Beckmann et al. (2009) . . . . . . . . . . . . . . . . . . . . . . 95

6.6 Group ICA statistical maps corresponding to the default mode

network and left and right frontoparietal networks are shown. The

statistical maps were thresholded using joint expected probability

distribution with height (p<0.001) and extent (p<0.001), corrected

at the whole brain level. . . . . . . . . . . . . . . . . . . . . . . . 99

6.7 Locations within the right-frontoparietal RSN (demarcated by red

boundary) which show significantly stronger connectivity for resistant

subjects compared to vulnerable subjects in baseline rested wakefulness

condition. Error bars show Mean ± SEM. . . . . . . . . . . . . . 100

6.8 Locations within the right-frontoparietal RSN (demarcated by red

boundary) which show significantly stronger connectivity for resistant

subjects compared to vulnerable subjects after sleep deprivation.

Error bars show Mean ± SEM. . . . . . . . . . . . . . . . . . . . 101

6.9 Locations within the right-frontoparietal RSN that show significant

decline in connectivity across the rested wakefullness and sleep

deprived states (RWSD), for all subjects. . . . . . . . . . . . . . . 102

A.1 Latencies in response for some of the flagship tablet devices. Courtesy

of Agawi TouchMarks, used with permission. . . . . . . . . . . . . 112

A.2 Screenshot of Quick PVT running on a Desktop system on Windows

and on a smartphone running Android. . . . . . . . . . . . . . . . 114

List of Tables

3.1 Standard parameter sets . . . . . . . . . . . . . . . . . . . . . . . 42

3.2 Estimation speed. The χ2 based estimator using random walk

simulation is the original method as proposed by Ratcliff et al.,

2011. As the estimation speed was prohibitively slow, we used the

proposed mixed simulation method with the χ2 based estimator

which while still slow, significantly improved speed. The proposed

MLE based estimator does not require any model simulation. . . . 44

3.3 Mean and standard deviation of error in estimation. Drift parameters

are normalized. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.1 Standard reaction time and diffusion parameter statistics of study

participants. All standard metrics were significantly affected by

sleep deprivation. In terms of diffusion parameters, except for

standard deviation in diffusion drift, all other parameters were

significantly affected. . . . . . . . . . . . . . . . . . . . . . . . . . 54

6.1 Clusters showing significant decline in connectivity in the right-

frontoparietal RSN across states (RW SD) for all subjects. . . . . 100

xxii

List of Abbreviation

BA Brodmann AreaBOLD Blood Oxygenation Level DependantCV Cross ValidationCRLB CramrRao Lower BoundDV Decision ValueDDM Drift Diffusion ModelDMN Default Mode NetworkEEG ElectroEncephaloGramESD Evening before Sleep DeprivationfMRI functional Magnetic Resonance ImagingGLM General Linear ModelsICA spatial-Independent Component AnalysisIID Independent and Identically DistributedISI Inter-Stimulus IntervalsKSS Karolinska Sleepiness ScaleMCMC Markov Chain Monte CarloMGF Moment Generating FunctionMLE Maximum Likelihood EstimateMNI Montreal Neurological InstituteNREM Non-Rapid Eye MovementPDF Probability Density FunctionPVT Psychomotor Vigilance TaskRBF Radial Basis FunctionRSN Resting State NetworkRT Reaction TimeRW Rested WakefulnessSD Sleep DeprivationSDT Signal Detection TheorySPM Statistical Parametric MapsSWA Slow-Wave ActivitySNR Signal to Noise RatioSVM Support Vector MachineTCFE Threshold Free Cluster EnhancementsVBLANK Vertical Blanking Interval

xxiii

List of Notation

∗ Convolution

[.]T Transpose∼ Distributed asE(X) Expectation with respect to Xvar(X) Variance of Xd(.)e Ceil function≈ converge in probability

xxiv

Abstract

Most of us have experienced sleep deprivation (SD) every now and then, and

know from experience that lack of sleep can adversely affect cognitive function

and vigilance. The neurophysiology of sleep and lack thereof, its association with

diseases and general well being has been studied extensively for over a century.

The lack of sleep has enormous economic, health and life cost. The loss of

performance and attention after SD can lead to industrial and transportation

accidents, medical errors and lapses in security. While the decline in performance

with SD is well established, it was recently (2004) observed that this decline in

performance varies significantly between subjects, with some subjects remaining

relatively unaffected while others show considerable decline in performance. This

subject specific vulnerability towards SD remains stable in time and shows trait

like features, i.e., the relative ranking of individuals according to subject specific

performance (on some behavioral task) is maintained over time irrespective of

sleep history. This suggests a stable neuro-cognitive basis for between-subjects

differences in performance. Being able to predict an individual’s vulnerability to

performance decline when sleep deprived is therefore of considerable interest.

The Psychomotor Vigilance Task (PVT) is a sustained-attention, simple one-

choice reaction time (RT) task that measures the speed with which subjects

respond to a visual stimulus. It is a proven assay for evaluating vigilance. We

use the PVT to evaluate and quantify the degradation of performance with SD

on three large independent data sets collected from two different labs over an

extended period of time. Instead of looking at the RTs using summary statistics

as it is traditionally done, we used the drift diffusion model (DDM) which is

a powerful model of perceptual decision making with strong neuro-behavioral

underpinnings. Using DDM we tried to address three fundamental questions: (1)

How are subjects vulnerable to SD differentially affected compared to resistant

subjects. (2) Can these differences measured prior to SD predict performance

xxv

following SD? (3) If there are measurable differences in DDM parameters between

vulnerable and resistant subjects, can these differences be supported and substantiated

by neuro-imaging experiments? In addressing these questions, we intended to gain

new insights into the neuro-behavioural underpinnings of differential vulnerability

to SD and to construct a classification system that can use easily measurable

behavioural data collected prior to SD to predict an individual’s vulnerability to

performance decline when sleep deprived.

To be able to reliably and efficiently use the DDM to address our research

objectives, considerable improvements had to be made to the DDM which was

only recently (Ratcliff, 2011) adapted to the one choice RT task like PVT. The

statistical properties of the model were poorly understood. The model further

lacked efficient ways to simulate and estimate the model parameters. Furthermore,

even if we assume that there are measurable differences between the vulnerable

and resistant subjects, it remains to be seen as to how these differences are

influenced by experimental conditions. Given the importance of wearable devices

and smartphones, the behavioural data may soon come from these portable devices

instead of controlled laboratory environments. Therefore, the performance of our

model must be ascertained across varying experimental conditions.

In this thesis, we made significant improvements in simulation and estimation

of the model and understood the statistical limitations of the model. Using the

DDM, we showed that the vulnerable subjects had a significantly slower rate of

information uptake compared to resistant ones even prior to any SD. This was not

anticipated by merely observing PVT performance. We tied these observations

to actual brain functions using functional magnetic resonance imaging (fMRI)

obtained from subjects in rested wakefulness prior to SD. Finally, we showed the

DDM parameters are the most important metrics when it comes to characterizing

vulnerability amongst the behavioral metrics. We constructed a classifier that

was capable of predicting vulnerability using only behavioral data, taken prior to

SD, accurately across data sets taken under varying experimental conditions. We

showed that the classification rates were reliable, reproducible and promising. Our

results will help clinicians gain a better understanding of differential vulnerability

to SD and the classification model has the potential to be used in many practical

settings.

Chapter 1

Introduction

We begin this chapter by providing a brief background on vulnerability to SD. We

discuss the driving force behind this research as well as the objectives and scope

of our work.

1.1 Motivation

The negative consequences of sleep deprivation (SD) are multifaceted and well

document in literature (Pilcher & Huffcutt, 1996; Durmer & Dinges, 2005; Knutson

et al., 2007; Grandner et al., 2010). Neurobehavioral effects include failure of

sustained attention (Lim & Dinges, 2008), reduced information processing capacity

(Kong et al., 2011, 2012), impaired memory consolidation (Diekelmann & Born,

2010), emotional dysregulation (Walker & van Der Helm, 2009), and altered

decision-making (Walker & van Der Helm, 2009; Vandekerckhove & Cluydts,

2010). The loss of performance due to SD has significant economic, health and

life cost (Colten et al., 2006). In many critical scenarios such as the military,

hospitals, transportation and security the ability to perform when sleep deprived

is a major part of work environment. Even minor differences in performance can

have major ramifications in such scenarios. As per the American Academy of

Sleep Medicine (AASM) an estimated 250,000 accidents every year (in the US)

are related to lack of sleep. The annual economic impact of accidents related to

SD is estimated to be between $43 to $56 billion (Leger, 1994). Williamson &

Feyer (2000), showed that even moderate levels of SD produces impairments in

cognitive and motor performance that is equivalent to legally prescribed levels

of alcohol intoxication. Work related sleep loss and fatigue in medical training

has become a source of increasing concern (Baldwin Jr & Daugherty, 2004). In

1

Chapter 1. Introduction

an intervention study, Landrigan et al. (2004); Lockley et al. (2004) showed that

medical interns who obtained 5 hours of less sleep, had 50% more attentional

failures, and committed 22% more serious errors on critical care units and made

5.6 times as many serious diagnostic errors while working a traditional schedule

compared with a schedule of reduced work hours. Similar results have been

obtained in studies of shift workers (Colquhoun, 1976; Folkard & Monk, 1979;

Richardson et al., 1989), truck drivers (Stoohs et al., 1994; McCartt et al., 2000;

Lyznicki et al., 1998), medical residents (Marcus & Loughlin, 1996; Steele et al.,

1999; Geer et al., 1997), and airline pilots (Neri et al., 2002; Price & Holley,

1989; Bourgeois-Bougrine et al., 2003). Apart from direct consequence of loss of

vigilance, SD also plays an important role in incidence of many diseases. SD

has been associated with increased all-cause mortality, adverse cardiovascular

outcomes, metabolic disorders, obesity and type 2 diabetes (Taheri et al., 2004;

Spiegel et al., 2005; Knutson et al., 2007). Gangwisch et al. (2006) showed that

SD is a significant risk factor for hypertension and it continued to be significant

even after controlling for obesity and diabetes. The progressive decline in sleep

duration over the decades as suggested by various polls across different countries

(National Sleep Foundation, US; Gallup, US; Great British Sleep Survey, UK;

Singapore Sleep Society, Singapore), adds to the importance of this study.

Sleep deprivation is thought to impair neurobehavioral functioning by destabilizing

performance, as evidenced by the capacity to perform well for short periods of

time, interrupted by occasional attention failures (Doran et al., 2001). The extent

to which sleep loss affects performance varies widely across individuals, with

some subjects remaining relatively unaffected while others show severe cognitive

impairment (Doran et al., 2001). Notably, between-subject differences in performance

are trait-like and stable across repeated exposures to sleep deprivation, irrespective

of sleep history (H. Van Dongen, Baynard, et al., 2004; H. Van Dongen, Maislin,

& Dinges, 2004; Lim et al., 2007). These findings lead to the concept of SD

vulnerability, and suggest that there is a stable neuro-cognitive basis for individual

differences in cognitive vulnerability to sleep loss.

Despite over a century of sleep research, vulnerability remains a relatively new

and poorly understood concept. In this thesis, we combine medical imaging,

statistical modeling and machine learning to gain insights into the science of

differential vulnerability to SD. Pharmaceutical companies, clinicians and neuroscientists

2


are keen to gain a better understanding of the mechanisms underlying vulnerability,

as it may shed new light into the understanding of sleep in general and the

plethora of diseases that associate with it. Being able to predict an individual‘s

relative performance decline when sleep deprived has important implications for

occupations that require long work hours and night shift work.

1.1.1 Wearable devices, Smartphones and Sleep

Slowly but surely smart phones are becoming ubiquitous. This is further accelerated

by rapidly dropping prices of smartphones especially on the AndroidTM(Google

Inc., Mountain View - California) platform. As of December of 2013, there

are already 1.43 billion smartphone users worldwide which is estimated to reach

2.5 billion in 2017 (Source: eMarketer Research, NY USA, Figure 1.1B). These

devices are also becoming portable and distributed data acquisition systems. So

much so that an entire industry of wearable devices that work in conjunction

with smartphones have come up (Figure 1.1A). This includes devices capable of

measuring heart rate, skin temperature, perspiration, calories burned, sleep time

and quality across the day. A strong motivation for the current work is to leverage

these technologies in the future to predict sleep vulnerability on mass scale and

provide unique insights into sleep dynamics above and beyond what is possible

currently.

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2012 2013 2014 2015 2016 2017

Use

rs

Year

Smartphone Users (Billions)

A B

Figure 1.1: A: Most popular wearable biosensor devices as of 2014. Nike, Fitbitand Jawbone combined constitute 97% market share. B: Number of smartphoneusers across time. The dotted lines are projections.

3


1.2 Research statement, scope and objectives

The main aim of this research is to develop a framework that provides a unifying

picture of vulnerability to SD. We intend to further our understanding of how

vulnerable subjects differ from resistant subjects, the underlying neuronal architectures

that mediate these differences, and finally, to predict vulnerability reliably and

efficiently using only rested data collected prior to any SD. One of the most

sensitive measures of performance impairment by sleep deprivation is vigilance

(Lim & Dinges, 2010), commonly assessed using the Psychomotor Vigilance Task

(PVT) (Dinges & Powell, 1985). During the PVT, participants maintain their

fastest possible reaction time (RT) to a simple visual stimulus presented at random

inter-stimulus intervals (ISI) ranging from 2 to 10 seconds. Traditionally, the RTs

are analysed using summary statistics like mean and median. Unfortunately, due

to nature of the RT distribution these summary statistics fail to capture the full

distribution. More importantly, these metrics are purely descriptive in nature

and lack any real biological basis and provide little insights into the mechanics of

the underlying process. To achieve our goals, we move beyond the traditional

ways of behavioral and clinical Psychology and apply the latest advances in

statistical modeling and machine learning. Specifically, we used the Ratcliff drift

diffusion model (Ratcliff & Van Dongen, 2011) (DDM) which is a powerful model

of perceptual decision making for single choice RT experiments like the PVT. The

DDM allowed us to understand the baseline differences between resistant and

vulnerable subjects in terms of the underlying cognitive processes. We conducted

functional Magnetic Resonance Imaging (fMRI) studies conducted on the resting

brain (i.e. in the absence of any task) to verify if the findings from DDM analysis

could be substantiated by independent imaging analysis. Finally, we constructed a

classification system incorporating all our prior knowledge to reliably and cheaply

predict the subject specific vulnerability to SD using only rested RT data derived

prior to SD.

To carry out our investigation, the DDM as proposed by Ratcliff et al., 2011

was substantially improved. The statistical properties, closed form solution and

efficient methods for estimating and simulating the model were developed. The

resting state fMRI analysis were carried out using model free, data driven methods.

Generalized linear models were extensively used to test hypothesis. All studies

were conducted within the context of total SD using the PVT as a measure

4


of vigilance. It should be remembered that while vigilance decrements are an

important and robust measure of performance decline in sleep deprived persons,

other cognitive domains may not be similarly affected. Therefore our results must

be interpreted within this context.

1.3 Issues and Challenges

The strongly multidisciplinary nature of our work presents some unique issues and

challenges. The present work can be divided into three rough categories:

(i) Model Improvement: This includes understanding the statistical properties

and limitations of the DDM, developing fast and efficient methods to simulate

and estimate the parameters of the model.

(ii) Design of Experiments: Clinical trials involving humans are not only

costly but also time consuming. Therefore it was of paramount importance

that the experiments including the hypothesis we wanted to test were well

thought of before hand. We collaborated with multiple laboratories working

in the forefront of sleep research to access data and resources that best

suited our research goals. Some of the clinical trials were guided by our

experimental results and were conducted longitudinally overs months. Ideally,

we would like to move away from expensive and time consuming albeit

controlled laboratory settings to cheap but unconstrained mobile platforms

and wearable devices that can be deployed in field conditions. To be able

to do so, the model’s predictive capabilities must be tested under widely

different experimental conditions. This also necessitates the development of

software solutions that would allow inexpensive and cross platform acquisition

of PVT data. Currently, such solutions exit, but are very expensive and only

available on limited platforms.

(iii) Imaging Analysis: Our investigation involves fMRI scans obtained under

rested condition in the absence of any task. Resting state fMRI is a relatively

new and powerful method for evaluating regional interactions that occur

when a subject is not performing an explicit task. In the absence of explicit

task, the 4D space-time data obtained from fMRI is essentially unconstrained

making it significantly more difficult to find patters from. Despite these

5


challenges, the major advantage of resting state fMRI is that if any patterns

associated with vulnerability to SD is found, especially before any incidence

of SD, then that would imply that these differences are intrinsic properties

of the brain mediated possibly by stable functional or structural differences.

Combined with the DDM analysis, it provides a powerful framework to

understand SD vulnerability.

1.4 Outline

Chapter 1 introduces the basic motivation and idea behind this research work

along with the scope and objective. Chapter 2 provides literature review of

some of related works in this field. This includes two separate sections, one

discussing the advances in the understanding of general and differential decline of

performance with SD. And the other discussing the advances in the modeling of

simple decision process. This includes an introduction to the DDM. In chapter 3,

the statistical analysis and improvements in the DDM are discussed. In chapter

4 we lay out the primary questions laid out in our report. We test our hypothesis

to see if DDM parameters are capable of discriminating vulnerable subjects from

resistant ones in the rested condition. We discuss the possible implications of our

results. In chapter 6, we discuss the imaging analysis. A brief introduction of

the issues and challenges concerning resting state imaging studies are presented.

We interpret the results within the context of DDM. In chapter 5, we construct

a classifier capable of predicting the vulnerability to SD from rested RT data.

The most important measures derived from baseline PVT performance that can

reliably predict vulnerability to SD are reported. In chapter 7, we summarize

the work in this thesis and provide a conclusion to the same. We also provide

directions for future extensions of the work. In Appendix A, we discuss currently

available software and hardware solutions to acquire PVT data. We also discuss

the technical challenges in developing a cheap and widely deployable software

based PVT acquisition system that can be used in conjunction with the DDM to

predict vulnerability on a mass scale. Finally, such a system is developed with

the hope that it would be instrumental in taking the current work further. As

the chronological ordering of references are important, we have followed American

Psychological Association (APA) reference style guidelines in this thesis.

6

Chapter 2

Review of Literature

The review of literature related to this study has been organized into four sections.

In the first section, we discuss the existing literature on performance decline with

SD, the neurophysiological models of sleep regulation and vulnerability to SD.

One of the most sensitive measures of performance impairment by SD is vigilance

(Lim & Dinges, 2010), assessed using mental chronometry (i.e., using RT tasks

like PVT). Therefore, in the second section, we discuss the standard way RT

distributions are studied. In the third section, we discuss the neurobiological basis

of simple decision making and models based on the underlying principles. And in

the forth section, we discuss the DDM and its advantages over other models. To

maintain a fluid presentation, the literature review for the imaging study, which

runs tangential to the current discourse, is discussed in Chapter 6.

2.1 Sleep Deprivation and Vulnerability

2.1.1 Quantifying Performance: The PVT

Degradation of vigilance is probably the most robust alteration of neurobehavioral

performance in healthy, sleep-deprived young adults (Lim & Dinges, 2010). The

Psychomotor Vigilance Task or PVT proposed by Dinges et al. in 1985, is a proven

assay for evaluating vigilance. The PVT is particularly attractive because of ease

of scoring, simple metrics and convergent validity (Dinges & Powell, 1985). The

test is short, free of learning effects (H. P. Van Dongen et al., 2003) and aptitude

difference making it suitable to use on subjects from widely different backgrounds.

Its simplicity makes it attractive for mathematically modeling how performance

fluctuates according to time-of-day. It is a sustained-attention, RT task that

measures the speed with which subjects respond to a visual stimulus. Standard

7

Chapter 2. Review of Literature

PVT runs for 10 minutes and the subject is shown a display terminal on which a

light appears randomly and a RT counter starts running to which the subject has

to respond as soon as possible via a hand held switch. The interval after which a

visual stimulus is shown is known as Inter-Stimulus Interval (ISI). In the standard

PVT, ISI is distributed uniformly between 2 to 10 seconds. The standard PVT

typically results in 95 responses. Figure 2.1 shows a subject taking the PVT using

a portable PVT monitor.

There are many variants of the standard PVT. For instance there is a shorter

5 minute variant of PVT which is shown to be viable alternative to the 10 minute

test (Loh et al., 2004), there are palm held PVT devices (Thorne et al., 2005),

there are instances especially in imaging studies where subjects take the PVT

lying down inside a Magnetic Resonance Imaging machine. PVTs longer than 10

minutes are uncommon, this is because longer-duration PVTs are more affected by

time-on-task effects (H. Van Dongen et al., 2011). While all variants of PVT are

valid it makes comparing results from different set of studies difficult. Therefore,

the experimental conditions under which PVTs are taken, play an important role

in our investigation.

2.1.2 Performance Degradation with SD

Early investigators studying the effects of SD assumed that exposure to extended

periods of SD would successively diminish the ability to perform neurobehavioral

tasks. The earliest scientific study of degrading effects of SD could be cited to

Patrick & Gilbert (1896). They reported that 90 hours of continuous wakefulness

caused both motor and cognitive deficits in three adults, although these findings

were not replicated in subsequent experiments (M. Smith, 1916; Robinson &

Herrmann, 1922; Robinson & Richardson-Robinson, 1922). Between 1923 to 1934,

Kleitman published a series of reports on SD that cleared these discrepancies. His

major finding (Kleitman, 1923) was that even after days of sleep deprivation

(60 to 114 hours), subjects could often transiently perform at baseline levels on

most tasks. There was therefore no evidence to suggest that SD could eliminate

specific motor and cognitive function. Instead, during SD most abilities could

be maximally utilized by a new effort. Further research by Warren & Clark

(1937) showed that SD affected neurobehavioral functions by initially destabilizing

performance rather than eliminating capacity to perform. Williams et al. (1959)

8


Figure 2.1: A subject taking the PVT in a reclined position. Inset: A portablePVT monitor, courtesy of artisan-scientific.com.

found that 78 hours of SD caused an 18-fold increase in the number of performance

lapses (defined as twice the subjects baseline mean) even though subjects were able

to perform fast and accurately throughout the 78 hours of waking. They further

observed that lapses were more likely to occur when electroencephalogram (EEG)

waveforms showed evidence of transition to sleep.

This observation led to so called lapse hypothesis which hypothesized that

performance during SD was affected by brief moments of low arousal during

which subjects were unable to respond. This meant that SD affected performance

variability rather than causing a smooth overall decline. The wake state instability

hypothesis (Doran et al., 2001) was proposed under similar lines of reasoning.

It hypothesizes that two competing forces are in act when sleep deprived the

homeostatic drive for sleep and compensatory effort to perform. When homeostatic

drive for sleep persists, RT slows down and lapses increase while when compensatory

efforts prevail, performance remains relatively unaffected. While appearing very

similar to lapse hypothesis there are important differences to be noted. The

9


lapse hypothesis suggests that performance during SD is essentially normal until

it is disrupted by brief periods of low arousal (or lapses). Contrary to this the

state instability hypothesis suggests that variability in neurocognitive performance

increases as homeostatic sleep-initiating mechanisms become progressively more

affected by SD. In other words, wake state instability occurs when sleep initiating

mechanisms repeatedly interfere with wakefulness and depending on the severity of

SD make cognitive performance increasingly variable and dependant of compensatory

mechanisms (see Figure 2.2). From a theoretical standpoint, what the wake state

instability suggests is that there are multiple, parallel and distinct mechanisms

by which the wake and sleep states interact giving rise to the overall behavioral

observations. The wake state instability hypothesis is now well established and

is consistent with the established model of sleep regulation (discussed in later

section). The interpretation of our results must be made within the context of

wake state instability and associated models of sleep regulation.

Figure 2.2: PVT RT in milli-seconds for a representative subject. Even after 12and 84 hours of SD, the subject was able to perform at baseline levels albeit, witha clear increase in the variability in performance. Courtesy of Doran et al. (2001).

2.1.3 Vulnerability to Sleep Deprivation

The decline of performance with SD is well established with over a century of

research backing it up. H. Van Dongen, Baynard, et al. (2004); H. Van Dongen,

Maislin, & Dinges (2004), showed that not only does the performance vary significantly

between subjects, but also that this subject specific difference is stable in time

10


showing a trait like feature. In simple terms, if a group of subjects are ranked

according to their performance on a specific behavioural task after SD and this

experiment is repeated in time, then the ranking will remain more or less same.

Furthermore, while the inter-individual differences are both significant and stable

across time, poor performance in one particular type of task does not imply poor

performance in another. Specifically, these trait like inter-individual differences

tend to cluster along three independent dimensions (see Figure 2.3): subjective

measures, objective measures of sustained attention and cognitive processing ability

(H. Van Dongen, Baynard, et al., 2004). In this investigation we are specifically

interested in the objective measures of SD, as they are of most practical value.

Therefore, care must be taken when extending the implications of our findings to

more general scenarios.

Mu et al. (2005); Lim et al. (2007), using imaging studies, showed that these

stable performance after SD are possibly mediated by stable neuronal architectures.

More importantly due to the intrinsic nature of these architectures, these behavioural

performance (E. Chua et al., 2014) as well as brain activation (Caldwell et al.,

2005; Y. L. Chuah et al., 2006; Chee et al., 2006) differences are observable

even prior to SD. Such findings raise the possibility that baseline measures can

be used to predict how well a person will perform when he/she is deprived

of sleep. It has been hypothesized, for example, that genetically-determined

differences in buildup and dissipation of sleep pressure, as assessed by the amount

of electroencephalogram slow wave activity in non-rapid eye movement (NREM)

sleep, can influence cognitive responses to sleep deprivation (Viola et al., 2007;

Goel et al., 2010; Bachmann et al., 2012; Bodenmann et al., 2012). It has yet to be

established, however, whether heritable differences in sleep structure contribute

to differences in performance during sleep loss. Nonetheless, it provides strong

case to construct a classifier to predict vulnerability using rested data. In the

next section, we will discuss about established model of sleep regulation, which is

important to understand the mechanics of sleep.

2.1.4 Model of Sleep Regulation

The two process model of sleep regulation proposed by Borbely (1982), posits that

sleep is regulated by the interaction of two processes: the homeostatic process S

and the circadian process C. The homeostatic process is responsible for the rise

11


Figure 2.3: Subjects were administered the Karolinska Sleepiness Scale (KSS-1), the Word Detection Task (WDT), and the Psychomotor Vigilance Task(PVT) after they underwent 24 hours of sleep deprivation (SD) on two separateoccasions. The KSS is a subjective measure of vigilance, the WDT is a test ofcognitive processing ability and the PVT is an objective measure of vigilance.The 21 subjects were arbitrarily labeled from A through U. Boxes demarcate firstexposure to SD and diamonds demarcate the second. While the ranking betweensubjects across repeated trials for all three tasks remain stable, there is no directcorrespondence between ordering in one task with ordering in another. Courtesyof H. Van Dongen, Baynard, et al. (2004).

of sleep propensity during waking and its dissipation during sleep. While the

circadian process is independent of prior sleep and waking, and is responsible for

the alternation of periods with high and low sleep propensity. This rhythm (which

is approximately 24 hours in healthy individuals) is generated by endogenous

neural oscillating systems entrained to the light-dark cycle by specific visual

pathways (Moore, 1983). This circadian clock or pacemaker synchronizes different

physiological rhythms, e.g. body temperature, melatonin secretion and cortisol

secretion.

12


Each of these processes are mediated via separate neuronal architectures and

to a large extent can be assumed to be independent of each other. In the

most simplest form, the interaction between the processes are assumed to be

linear. The time course of Process S is derived from the changes of EEG slow-

wave activity (SWA, EEG power in the 0.75-4.5 Hz range). The process C,

is independent of sleep and waking, modulates the thresholds which determine

the onset and termination of a sleep episode, respectively (see Figure 2.4). The

exact model equations have been derived from experiments and a compilation

can be found in Achermann & Borbely (2003). Based on refinements of the two

process model, many variations have been proposed (see Achermann & Borbely

(2003) for an overview). Within the context of our investigation, the two process

model in its simplest form provides a powerful conceptual framework for the

analysis of experimental data and results without overcomplicating the analysis

with irrelevant technicalities.

Figure 2.4: Two-process model of sleep regulation. Process S indicates thehomeostatic built-up of sleep pressure and Process C represents the circadianrhythm. The difference between the two processes quantify the sleep pressure.

2.2 Mental Chronometry

Mental chronometry is the use of RTs in perceptual-motor tasks to infer the

content, duration, and temporal sequencing of cognitive operations and it is the

13


core paradigm of experimental and cognitive psychology. For our investigation,

PVTs are the primary behavioural task to quantify vigilance. The end result of

such an experiment is a set of RTs, which must be analyzed further. In the next

section, we will discuss the standard ways RTs are processed, their shortcomings

and possible solutions.

2.2.1 Standard RT Analysis

RTs have been used for a long time in cognitive psychology to measure behavioral

performance (Luce, 1986). To draw inferences about mental processes researches

initially relied upon summary statistics like mean, median and standard deviation

of RT. It was soon realized that these measures fail to capture the complete

RT distribution. RT distributions are notoriously skewed (Figure 2.5) and more

often than not censored (due to limitation of time). Ignoring the RT distribution

shape can have detrimental effect on the analysis. To address these issues, early

researchers tried trimming the data or apply non-linear transformations (like

reciprocal transformation) to normalize the RT distribution. While this makes

the data more symmetrical but this is at the cost of throwing away or ignoring

important aspects of the tail. Using higher order moments like skewness and

kurtosis to capture full information contained in the RT distribution has severe

drawbacks (Ratcliff, 1979). Sampling variances associated with higher order

moments is large, making it difficult to get good estimates with practical size

of RT samples. Furthermore, higher order moments are extremely sensitive to

outliers. An alternative approach is to explicitly assume a model for the shape of

RT distribution. Soon focus shifted to parameterized statistical models to describe

the distribution. In the next section we will discuss some of the popular models.

2.2.2 Statistical Model of RT distribution

The advantage of parameterized statistical models is that they capture the full

RT distribution. Many statistical models were proposed like gamma (McGill &

Gibbon, 1965) and log-normal distributions (Luce, 1986) but they are not popular

and do not provide good enough fit. Hohle (1965), proposed a decomposition of RT

into decision and residual component. He assumed that the decision component

has exponential distribution and the residual has normal distribution and both the

components are independent. This results in an ex-Gaussian distribution which

14


Figure 2.5: A sample reaction time (RT) distribution. RT distributions tend tobe strongly skewed towards right.

is convolution of exponential and normal distribution. It has three parameters

µ, σ and τ , where µ and σ are the mean and standard deviation of normal

component and τ parameterizes the exponential component. The expression for

the ex-Gaussian is:

f(t) =(e

−(t−µ)τ

+ σ2

2r2 )

(τ√

2π)

∫ (t−µ)σ−στ

−∞e−y

2/2dy (2.1)

The assumption made by Hohle remains unproven, although, it was very successful

as a convenient summary of empirical RT distribution (Heathcote et al., 1991;

Ratcliff, 1988).

While these models provide a good summary of the RT distribution and are

certainly better than using simple central tendencies or summary statistics they

still have a major downside; these models are purely descriptive, they lack any

neurobiological basis. In the absence of a theoretical description behind the

model, it is not possible to tie the parameters of the model to the underlying

cognitive process and the parameters of the model are in some sense, arbitrary.

Therefore models inspired by neurobiology of decision process are desirable. In

the next section, we will discuss mathematical models of perceptual decision

making which are based on the neural basis of decision making as observed with

experimentations.

15


2.3 Models of Perceptual Decision Making

2.3.1 Signal Detection Theory

Signal detection theory (SDT) is one of the most successful formalisms ever used to

study perception (Gold & Shadlen, 2007). SDT provides a framework to convert

a single observation of noisy evidence into a categorical choice. Most sensory

motor tasks can be thought of as a form of statistical inference (see Parker &

Newsome (1998) for a review). Therefore SDT is the natural framework to study

them. It starts with the basic assumption that all decision process takes place

in the presence of some uncertainty. SDT provides a mathematical structure to

this notion. A single observation of noisy sensory evidence e is gathered based

on which a particular hypothesis h1 to hn, n >= 1 is selected or rejected. The

evidence e is derived from the senses and might be the spike count from a single

or group of neurons or it might be the difference in spike counts between groups of

neurons. It is important to note that e is caused by a stimulus controlled by the

experiment and is regarded as a random variable as it is corrupted by noise. The

decision process requires the construction of decision value (DV) from the evidence

e. SDT provides a flexible framework in that it allows formation of decisions that

integrate priors P (hi), evidence and subjective costs and benefits associated with

each of the potential outcomes. Accumulator models which rely on sequential

analysis are a natural extension to SDT that accommodates multiple pieces of

evidence in time. In these models the noisy evidence is accumulated and the DV

is updated with new evidence until it is sufficient to make a decision.

2.3.2 Accumulator Models

Sequential analysis differs from SDT in one important aspect; it assumes that the

decision has two parts, the usual one between hypothesis h1 to hn and another that

decides whether or not it is time to stop the process and commit to a particular

hypothesis. Figure 2.6 illustrates the framework. The decision process relies on

a sequence of observation in time. After each acquisition, the DV is computed

using some function fi(e) using evidence ei obtained up to that point, now the

process may continue, in which case the accumulated evidence is carried forward

or a decision can be made based on the value of DV at that point. In this context

e0 can be thought of as prior probability of the hypothesis. Notice that both

16


the criteria that stop the process as well as the function fi which converts the

accumulated evidence to a DV could be dynamic and vary with time. Based on

different stopping criteria ci and DV functions fi we can get different accumulator

models. We are interested in a specific kind of accumulator model called the drift

diffusion model (DDM) which assumes that the evidence is sampled from a normal

distribution in infinitesimal time steps with the process being stopped when the

accumulated evidence exceeds a fixed positive or negative number. The model

is highly successful in modeling most two choice decision processes and recently

adapted to a single choice task like the PVT. Before we discuss the DDM, we will

discuss some of the experiments that show the neural basis of perceptual decision

making. The accumulator model in general and DDM in particular are derived

from such experiments.

Figure 2.6: General framework of sequential analysis model.

2.3.3 Perceptual Decision Making: Neural Basis

The first series of experiments investigating decision making were conducted on

primates. Single unit recording studies in monkeys show causal link between

behaviour and the activity of neuronal population in sensory regions (Shadlen

et al., 1996; Salinas et al., 2000; Gold & Shadlen, 2002; Romo & Salinas, 2003).

Similar studies (H. Heekeren et al., 2004; Pleger et al., 2006; Kaiser et al., 2007)

have been conducted on humans. These studies not only demonstrate that the

representation of evidence in sensory regions is used to make perceptual decisions

17


but also provide detailed knowledge about the nature of these representations.

These experiments strongly support the accumulator model of decision making

- sensory evidence is integrated and decision variable is constructed out of it.

These studies show that the neural activity (spike count) in areas involved in

decision making, increases gradually and saturates until a response is made. More

importantly the rate of information is directly proportional to the difficulty of the

trial. Particularly, they make way for the DDM (see Figure 2.7). Although,

neurophysiological studies show that neural systems not involved in the decision

process (like the motor systems) are involved in selecting and planning certain

actions which are important from the perspective of decision making. Therefore,

the process of decision making is not strictly serial. Notwithstanding the fact

that the accumulator model is only an approximation, the DDM still performs

very well and is now a well established model of perceptual decision making with

multiple experiments supporting it. In the next section we will discuss the DDM

in broader detail.

2.4 The Drift Diffusion Model

As discussed in the previous section, the DDM is inspired by experiments on

primates and humans and is a good approximation of the underlying neurobiology

of decision making. The DDM is based on the diffusion process which is a

stochastic process that develops through time. The diffusion process was first used

in the context of memory retrieval by Ratcliff in his seminal 1978 paper. A Markov

process in which continuous change of state space occurs is called a diffusion

process. If the process is characterized by the random variable X (t) ; t ≥ 0,

denoting the position of the process in state space at time t, then the process

can be studied using transition probability densities p(x0, t0;x, t) which give the

probability density function (pdf) of X(t) as a function of x conditional on

X (t0) = x0. In other words,

prob (a < X (t) < b | X (t0) = x0) =

∫ b

a

p (x0, t0;x, t) dx (t0 < t) (2.2)

The variable x0 and t0 are the backward variables since they refer to earlier

time and x and t are called forward variables following similar reasoning. By

18


Figure 2.7: Top: The accumulator model assumes that the perceptual decisionmaking is a serial process progressing from perception to action. Bottom:Diffusion model is an accumulator model for the neurobiology of decision makinginspired by experiments. The figure shows the model for a two choice visualtask (house or face?). It assumes that decisions are formed by continuouslyaccumulating sensory information until one of the response criteria are met (a or-b). The rate of information accumulation µ, models the firing rate of the neuronsinvolved in the decision process. The rate is lower (red path) for a difficult taskwith low sensory evidence and higher (green path) for a easy task (high sensoryinformation). The moment to moment fluctuations in the sample path reflect thenoise in the decision process. Figure adapted from H. Heekeren et al. (2004).

virtue of Markov property, p(x0, t0;x, t) will satisfy the Chapman-Kolmogorov

equation. The Chapman–Kolmogorov equation (Gardiner et al., 1985) is an

identity relating the joint probability distributions of different sets of coordinates

on a stochastic process. In our case, p(x0, t0;x, t) will satisfy the forward and

backward Kolmogorov equations 2.3 and 2.4, where β(x, t) and α(x, t) are the

infinitesimal mean and variance of the process.

1

2

∂2

∂x2α (x, t) p − ∂

∂xβ (x, t) p =

∂p

∂t(2.3)

1

2α (x0, t0) p

∂2p

∂x20

− β(x0, t0)∂p

∂x0

= − ∂p∂t0

(2.4)

Based on the stopping criteria boundary conditions are then imposed on the

19


stochastic differential equation. For example for a two hypothesis (or two choice)

task we can assume that the process starts at some point z and the boundaries

are at y = 0 and y = a (Figure 2.8, right) then the boundary conditions are:

p (z, 0;x, 0) = δ (x− z) ; p (z, 0; a, t) = 0; p (z, 0; 0, t) = 0 (2.5)

Similarly for a single hypothesis (one choice) task we assume that the process

starts at 0 and the boundary is at y = a (Figure 2.8, left). Then the boundary

conditions are:

p (0, 0; a, t) = 0; p (0, 0;x, 0) = δ(x) (2.6)

The function δ(x) is the Dirac delta function, a degenerate probability function

with all its mass at 0. At the boundary itself density must be equal to 0 so that the

terminated process disappears from transition density. When β and α are fixed,

which is what we are interested in, Equation 2.4 along with boundary conditions

(Equation 2.5 or 2.6) describes a diffusion process with constant drift and constant

variance with absorbing boundaries. If β = µ and α = σ2 then such a process

is equivalent to a Wiener process (or standard Brownian motion) with drift µ

and variance σ2. The easiest way to theoretically handle such diffusion process

is to think of it as a continuous limit of simple random walk. This approach

gives considerable insights into the continuous diffusion process and in most cases

allows us to find a complete probabilistic description of the continuous process by

studying the system under the limiting condition of zero time intervals or ∆t→ 0.

This approximation is very useful when a closed form solution is not available.

Figure 2.8: A diffusion process with constant mean β and variance α is equivalentto a Wiener process. Shown here are sample paths for such a diffusion processmodeling task with one and two hypothesis respectively.

20


2.4.1 Diffusion Process Applied to Perceptual DecisionMaking

Onset of stimulus

Decision process starts

Decision process

stops

Response

t1 = pre-decision time, t2 = post decision time

Ter = t1+ t2

RT = Td+Ter

Tdt1 t2

Time

Figure 2.9: Components of reaction time.

The diffusion process models only the decision part of the accumulator model.

The diffusion process, when applied to behavioral tasks has additional complexities

(Ratcliff & Rouder, 1998). The decision process is broken down into three parts.

From the onset of stimulus, it takes some time t1 to encode the stimulus after

which the decision process starts which is modeled using diffusion process. Once

the boundary is hit decision process stops after which some more time t2 is required

to decode this decision and act. If the total decision time is Td then the reaction

time (RT) is simply the sum of all times spent in encoding, decoding and decision

(Figure 2.9). t1 and t2 are normally combined and referred to as non-decision time

Ter.

Reaction T ime = t1 + Td + t2 = Td + Ter (2.7)

For a two choice (two hypothesis) tasks there are two decision boundaries,

without loss of generality one boundary is assumed to be at 0. This gives rise

to two parameters, the starting point z0 and the other boundary a. The starting

point z0 is normally expressed in terms of boundary separation a and encodes the

relative value or bias of one hypothesis with respect to other. For instance a z0

value of a/2 suggests an unbiased attempt at the task. The drift rate µ and σ2 are

parameters of diffusion model. It turns out that σ2 is simply a scaling parameter,

in other words, if σ is doubled the other parameters of the model can be doubled to

get exactly same model. The choice of σ is arbitrary but we will fix its value at 0.1

in accordance with Ratcliff for ease of comparisons. While for a single trial there

21


are four parameters µ, z0, Ter and a, when studying subjects some or all of these

parameters can be allowed to vary across trial. While variability in drift rates is

expected as subjects do not encode stimulus in exactly the same way every time,

variability in other parameters may not be so intuitive. A simple diffusion model

where parameters do not vary fails to fit experimental RT distributions. Moreover,

this affects estimation of parameters adversely by introducing large biases and

standard deviation. The RT distributions obtained from experiments can be

thought of as contaminated by alternate processes like fast guesses, distractions,

changes in motivation and simple RT outliers. The variability introduced in the

parameters is simply an implicit way of modeling these contaminants. Variability

in boundary a is observed to be of least significance while variability in drift is

absolutely necessary to explain experimentally obtained RT distributions. The

effects of parameter variability on estimators are discussed at length in (Ratcliff

& Tuerlinckx, 2002). Most frequently used two-choice RT task diffusion model

has seven parameters:

(i) Mean drift rate: ξ

(ii) Across trial variability in drift rate: η

(iii) Boundary separation: a

(iv) Mean starting point: z

(v) Across trial variability in starting point: sz

(vi) Mean non-decision time: Ter

(vii) Across trial variability in non-decision time: st

Drift is assumed to have a normal distribution N(ξ, η2) while other parameters

have a uniform distribution. The model is illustrated in Figure 2.10. These seven

parameters constitute the parameter vector Θ ∈ R7 such that,

Θ = [ξ, η, a, z, sz, Ter, st]T (2.8)

where [.]T is the transpose operator.

22


Figure 2.10: Seven parameter diffusion model for two choice RT task. The non-decision time and the starting point are assumed to be distributed uniform. Thedrift varies from trial to trial according to a normal distribution.

2.4.2 Extending DDM to The PVT

The PVT, which we will be using extensively in this investigation is a one choice

task. The DDM was recently adapted to one choice RT tasks (Ratcliff & Van Dongen,

2011). In the one choice model there is only one boundary. Without loss of

generality the process is assumed to start from 0 and the boundary is placed at

a. As a result of this the number of parameters reduce to five, [ξ, η, a, Ter, st]T .

The model is illustrated in Figure 2.11. With these five parameters the model is

not uniquely identifiable. If the value of a is doubled ξ and η can be doubled to

give the same distribution of RTs. Due to the difficulty in uniquely identifying

model parameters in single boundary tasks (Ratcliff & Van Dongen, 2011), we

followed Ratcliffs suggestion to use uniquely identifiable parameter ratios (ξ/a

and η/a). This resulted in a model with four parameters Θ = [ξ/a, η/a, Ter, st]T .

In other words, the drift parameters of one subject cannot be compared with

another without first normalizing their estimated boundary values. For the sake of

simplicity, from this point onwards we will use drift and drift ratio interchangeably.

When comparing the drift parameters across group or across subjects, it is implicit

23


that they were normalized by the estimated boundary parameter. Methods for

simulating and estimating parameters of the one choice DDM will be discussed

in detail in Chapter 3. As we are using the DDM in the context of PVT, unless

specified otherwise, DDM refers to the one-choice DDM.

Time

Evidence criterionɑ

Ter

St

high drift

low drift

mean drift ξ

SD(across trial) η

Figure 2.11: Five parameter diffusion model for one choice RT task. The non-decision time is assumed to be distributed uniform. The drift varies from trial totrial according to a normal distribution.

2.4.3 Experiments Supporting The DDM

An attractive feature of diffusion modeling is that it can predict the response

time distribution under different contexts (Ratcliff, 2002) and varying levels of

noise (Ratcliff & Tuerlinckx, 2002). The model has been tested by manipulating

various facets of the decision process and then observing the corresponding change

in diffusion parameters (Voss et al., 2004). They used a color discrimination task

to investigate how changing specific aspects of decision process had effects on the

corresponding parameters in diffusion model analysis. They found that decision

thresholds were higher when subjects were motivated towards higher accuracy,

drift rates were lower when stimulus were harder to discriminate. The ability of the

diffusion process to separate and identify the different components of the decision

24


process allowing fine-grained interpretations is what makes it so attractive. For

instance Ratcliff et al. (2003) using the diffusion model to study the effect of aging

on brightness discrimination task came to the remarkable conclusion that older

subjects perform just as good as young subjects in all aspects of performance,

except those directly involved in executing the decision (parameterized by Ter).

Among the numerous studies using the PVT to characterize performance

in sleep-deprived persons, there exists only one (Ratcliff & Van Dongen, 2011)

that used diffusion modeling to explain behavior. Diffusion parameters in the

SD condition were compared to those obtained in the non-deprived condition.

Although SD was found to negatively affect several diffusion model parameters,

only the effect on the drift (ξ) parameter was statistically significant. How DDM

parameters are differentially affected between subjects based on their vulnerability

to SD is not yet know. Figure 2.12, provides a bird‘s eye view of the time-line of

the most important findings pertaining to our research.

2.5 Summary

In this investigation, we want to improve the understanding of differential vulnerability

to SD. This requires integration of different fields including clinical and behavioural

psychology, neuroscience, statistics and machine learning. In this chapter, we

have reviewed the literature that provides broad context to the work we are

doing. We discussed the shortcomings of standard RT measures and earlier models

of perceptual decision making compared to the DDM. Some of the literature

which are essential in our analysis but do not directly relate to our objectives

are discussed in their respective chapters.

25


1896

1923

1934

1937

1959

1965

1978

1982

1985

1998

2001

2004

2006

2007

2010

2012

2013

2011

time

Patrick & Gilbert

First reported scientific study of SD

Kleitman et al.SD causes variable

performance by destabilizing cognitive

and motor skillsWarren & Clark

William et al. Lapse Hypothesis

BorbelyTwo process model of

sleep regulation

Dinges et al. PVT as a measure of vigilance

HohleEx-Gaussian

RatcliffTheory of memory

retrieval – Diffusion process

Ratcliff et al.Two choice DDM

Doran et al. Wake state instability hypothesis

Van Dongen et al. Concept of vulnerability to SD

Ratcliff et al. One choice DDM

Bachmann et al., Bodenmann et al.

Viola et al.

Goel et al.

2005

Caldwell et al.

Chuah et al., Chee et al.

Chua et al.

Genetic markers of vulnerability to SD

Baseline difference in brain activations between vulnerable and resistant

group

Baseline difference in behavioral performance between vulnerable and

resistant group

Not to scale

Figure 2.12: Timeline shows the major findings in the context of our investigation.While all findings are important, the ones marked in red are heavily relied on inthe current work.

26

Chapter 3

The One Choice DDM:Simulation and Estimation

Despite the popularity of one choice RT tasks (like the PVT) in cognitive

psychology, the DDM was only recently adapted (Ratcliff & Van Dongen, 2011)

to one choice tasks. The wide spread use of the model is restricted due to

complexity involved in estimating the model parameters from data. Furthermore

the current method for simulating the model is prohibitively slow. We propose

improvements in the simulation process that considerably improves runtime. We

also derive a closed form solution for the model and propose a maximum likelihood

estimator to determine the parameters. In the next section we will discuss the

statistical properties of the model, its limitation, existing methods for simulating

and estimating the model and improvements on it. The current analysis is not

only important for the current investigation but also essential for the widespread

use of one choice DDM in the clinical community.

3.1 Approximate the Diffusion Process

The closed form solution for the DDM has not yet been derived. In the absence

of a closed form solution, the easiest way to generate samples from the model is

to approximate the diffusion process by a simple random walk. This is done by

discretizing the time and evidence axes into small steps. In a simple random walk

The analysis presented in this chapter appears partly in Patanaik, A., Zagorodnov, V.and Kwoh, C. K. “Parameter estimation and simulation for one-choice Ratcliff diffusionmodel.”, Symposium on Applied Computing-Association for Computing Machinery (SAC-ACM) Special Interest Group on Applied Computing (SIGAPP), 24-28 March 2014. doi:10.1145/2554850.2554872. Used with permission.

27

Chapter 3. The One Choice DDM: Simulation and Estimation

successive one step transitions are allowed only to neighboring states only. If the

step size is ∆ taking place in small time intervals τ , then at the limiting case

when τ → 0 we will get the diffusion process. Without loss of generality let us

consider a particle at the origin, after each time step τ , it takes a step Z where

the probability of taking a positive step is p and probability of taking negative

step is q, then

prob (Z = +∆) = p (3.1)

prob (Z = −∆) = q, (3.2)

Because all steps are assumed to be mutually independent, the moment generating

function (mgf) of one step can be easily computed as,

E(e−θZ

)= pe−θ∆ + qeθ∆ (3.3)

After time t there will be t/τ steps resulting into total displacement of X(t),

which is the sum of the n independent random steps each with mgf described in

3.3. Therefore, mgf of X (t) is

E(e−θX(t)

)=(pe−θ∆ + qeθ∆

)t/τ(3.4)

The mean and variance of X(t) are

E (X (t)) =

(t

τ

)(p− q) ∆, V (X (t)) =

(t

τ

)(4pq) ∆2 (3.5)

Under limiting conditions as τ → 0 and ∆ → 0 the mean and variance of

the simple random walk should converge to drift and variance parameter of the

diffusion process. That is

(t

τ

)(p− q) ∆→ µ ∗ t and

(t

τ

)(4pq) ∆2 → tσ2 (3.6)

These limits will be satisfied when

∆ = σ√τ , p =

1

2

(1 +

µ√τ

σ

)(3.7)

Note that σ here is the variability in the diffusion process (not the across trial

variability in drift η in the DDM) and as discussed in Section 2.4.1 acts as a

scaling parameter for the DDM.

28


3.2 Simulating the DDM

Equation 3.7 ties the diffusion model to random walk. By using small values of

τ , the diffusion process can be approximated. The choice of τ parameter trades

off with speed; using a very small τ will make the simulation more accurate

at the expense of increased computation time. Given a set of parameters Θ =

[ξ, η, a, Ter, st]T , the existing method to simulate the DDM detailed briefly by

Ratcliff & Van Dongen (2011) is to sample the drift parameter µ from the appropriate

normal distribution µ ∼ N(ξ, η2) and the non-decision time ter from appropriate

uniform distribution ter ∼ U(Ter, st). For the particular sampled drift µ the

diffusion process is simulated using the random walk approximation. The decision

time is then the time taken by the random walk process to hit the boundary a.

The sampled RT is simply the sum of sampled non-decision time and the decision

time from random walk approximation. On an average case, the complexity of

simulation is O( aξτ

). For practical purposes, with τ = 0.5ms, and a/ξ ≈ 250ms

which represents an average PVT sample, approximately 500 cycles are required

to generate a single simulated RT sample. This makes the simulation process

extremely slow, especially when a large number of samples are required. The

process is illustrated in Figure 3.1.

𝜃 = [𝜉, 𝜂, 𝑎, 𝑇𝑒𝑟, 𝑠𝑡]

Sample drift 𝜇 from normal distribution

𝑁(𝜉, 𝜂)

Sample non-decision time 𝑡𝑒𝑟 from uniform

distribution

𝑈(𝑇𝑒𝑟 −𝑠𝑡2, 𝑇𝑒𝑟 +

𝑠𝑡2)

Random walk simulation with drift 𝜇 and boundary 𝑎

Non-decision time 𝑡𝑒𝑟

Decision time 𝑡𝑑

+ Simulated RT

Figure 3.1: Figure illustrates the standard simulation process for the DDM.

3.2.1 A simplified version of DDM

If we assume that the diffusion parameters do not vary across trials then the model

can be considerably simplified. This can help in simplifying and considerably

improving the speed of DDM simulation. The analytical solution for RT distribution

29


can be obtained by solving the first passage time for a Wiener process X(t) with

drift µ and standard deviation σ with one absorbing boundary at x = a. For

simplicity the effect of Ter can be considered later, for now we can assume the time

axis shifted to Ter. The diffusion process will follow the Kolmogorov equations

(Equation 2.3 and 2.4) with α = σ2 and β = µ. Consider the forward equation

for now:1

2σ2 ∂

2p

∂x2− µ∂p

∂x=∂p

∂t(3.8)

Subject to the conditions:

p (0, 0;x, 0) = δ (x) , p (0, 0; a, t) = 0 (t > 0) (3.9)

These equations arise naturally in heat conduction and diffusion problems. Solving

the partial differential equations we obtain (see Cox & Miller (1977), page 221 for

details)

p (0, 0;x, t) =1

σ√

2πt

[e

− (x−µt)2

2σ2t

− e

2µa

σ2− (x−2a−µt)2

2σ2t

](3.10)

If T is the first passage time from 0 to a then the probability density of RT is

f(t). For µ > 0, the process is bound to reach the boundary at a and f(t) can

then be computed as:

f (t) = − d

dt

∫ a

−∞p(0, 0;x, t)dx

= − d

dt

φ

(a− µtσ√t

)− e

2µa

σ2 φ

(−a+ µt

σ√t

)=

a

σ√

2πt3e

− (a−µt)2

2σ2t

(3.11)

where φ is the standard normal integral. The mgf of T can be computed as:

f ∗(s) =

∫ ∞0

e−stf(t)dt = e[ aσ2

(µ−(µ2−√µ2+2sσ2))] (3.12)

Differentiating twice with respect to s and setting s to 0 give the first two moments

as:

E (RT ) =a

µV (RT ) =

aσ2

µ3(3.13)

Subtracting back Ter from equation 3.11 we obtain the distribution of RT as:

f (RT ) =a

σ√

2π(t− Ter)3e

− (a−µ(t−Ter))2

2σ2(t−Ter)

(3.14)

30


The mean and the variance of the RTs will be:

E (RT ) = Ter +a

µV (RT ) =

aσ2

µ3(3.15)

It is interesting to notice that the distribution of RT (equation 3.14) is actually

a shifted inverse Gaussian distribution. Its statistical properties are well studied

(Folks & Chhikara, 1978).

3.2.2 A mixed simulation method

While the equation 3.14 is valid for all values of drift µ, the process is guaranteed

to terminate only if the drift is positive. On the other hand if drift is zero or

negative (µ <= 0) then the process may never terminate and the distribution in

equation 3.14 becomes defective. In other words for negative or zero drift value,

the process has a finite probability of never terminating and the point mass at

infinity is:

P (T =∞) = 1− e2µ

σ2 (3.16)

When the drift is positive, samples from equation 3.14 can be generated (Michael

et al., 1976) using the algorithm 1.

Algorithm 1 Sampling from shifted inverse Gaussian distribution

Given distribution parameters: drift µ > 0, boundary a and shift terSample a random variable α from a standard normal distribution:α = N(0, 1)Sample another random variable β from a uniform distribution between 0 and1:β = U(0, 1)Now obtain the value of x as:

x = µ+ µ2σ2α2

2a2− µσ2α

2a2

√4µa2

σ2 + µ2α2

if β <= µµ+x

thenreturn x+ ter

elsereturn µ2

x+ ter

end if

Using this algorithm, samples for the DDM can be generated much more

efficiently and accurately when drift is positive without relying on random walk

approximation. For most experiments the proportion of negative or zero drift is

very small. The DDM simulation starts similar to previously described method;

given the parameters of the DDM, Θ = [ξ, η, a, Ter, st]T the drift parameter µ

31


is sampled from the appropriate normal distribution µ ∼ N(ξ, η2) and the non-

decision time ter from appropriate uniform distribution ter ∼ U(Ter, st). If the

sampled drift µ <= 0, then simulation is carried out as usual using random

walk simulation. But if the sampled drift µ > 0, then the appropriate shifted

inverse gaussian distribution is sampled using algorithm 1. By relying on the

shifted inverse Gaussian distribution for positive drifts and falling back on random

walk approximation otherwise can significantly improve simulation speed without

compromising accuracy. We refer to this method of simulation as the mixed

simulation method.

3.2.3 Simulating PVT

Once we can simulate the DDM, simulating PVT is straightforward. A standard

PVT runs for 10 minutes with the ISI varying uniformly between 2 and 10 seconds.

The maximum time allowed to respond is usually 10 seconds. To simulate the

PVT, we first sample uniformly between 2 and 10, this constitutes the simulated

ISI. Then a RT is simulated from the DDM. If the sampled RT > 10 then the RT

is taken to be 10. The sampled RT is added to the sampled ISI and the process is

continued till the total time exceeds 10 minutes. The sequence of RTs generated

constitutes a simulated sample of the PVT. Notice that as the duration of PVT

is fixed, slow responses can strongly affect the number of samples collected within

the time-frame of the test.

3.3 DDM parameter estimation

3.3.1 Problem statement

Given a set of N response times generated by a one choice task experiment (RT

data), such as PVT experiment, we assume that each response is independent and

identically distributed (IID) and is a result of an underlying DDM parameterized

by Θ = [ξ, η, a, Ter, st]T . The goal is to estimate these parameters as accurately

and efficiently as possible from the least amount of RT data. To be useful,

the estimation process must be fast, unbiased and should have low variance in

estimation error. Irrespective of the estimation algorithm, the efficiency (variance

in estimation error) cannot be lower than the CramrRao lower bound (CRLB),

which states that the variance of an unbiased estimator is at least as high as

32


the Fisher information (Radhakrishna Rao, 1945). This is important in our

investigation because it provides an upper bound on the performance of any

estimator for a given size of data.

3.3.2 DDM: identifiability issues

As discussed in Chapter 2, the five parameter DDM is not uniquely identifiable.

In the subsequent chapters, while estimating, all parameters are assumed to be

free, but when reporting results, invariant parameter ratios (ξ/a, η/a) are used

instead of the original parameters.

3.3.3 Current estimation method

The parameter estimation process briefly described by Ratcliff & Van Dongen

(2011), relies on the simulation based on random walk approximation and leaves

several free parameters that can be altered depending on the data. To estimate

the parameters Θ of the model, RT distribution is simulated and compared with

RT data obtained from experiments using a χ2 goodness of fit. An iterative

algorithm based on simplex minimization routine (Nelder & Mead, 1965) is used

to successively improve the model fit. For the simulation process 20,000 RTs per

distribution are obtained. A step size of 0.5 ms for the random walk approximation

is chosen. To fit the model to data, 0.05, 0.1, 0.90, 0.95 quantiles of the RT

distribution are obtained. These quantile RTs are used to find the proportion of

responses in the RT distribution from the model lying between the data quantiles

which are multiplied by the number of observations to get the expected values (E).

The observed values (O) are simply 0.05 multiplied by the number of observations.

A χ2 statistic is then computed as∑

(O−E)2

E. The method is very sensitive to

initial estimate of the parameters, therefore good starting point for the estimator

is essential to avoid local optimas. Ratcliff & Van Dongen (2011) suggested using a

Markov chain Monte Carlo (MCMC) based maximum likelihood estimate (MLE).

The MCMC-MLE method is discussed in next section. The estimation process is

summarized in Figure 3.2.

3.3.4 MCMC-MLE

To initialize the estimation process, Ratcliff & Van Dongen (2011) used MCMC-

MLE. In their paper, the method is not described in any detail. MCMC-MLE is a

33


DATA

MCMC-MLE

Nelder-Mead simplex algorithm

goodness of fit

χ2

good fit

bad fit

return Θ

simulated distribution

observed distribution

Observed RT

Θ0initialization update Θ

sample for i=1 to 20,000 iterations

Random walk simulation

Figure 3.2: Figure illustrates the DDM parameter estimation process as describedby Ratcliff & Van Dongen (2011).

combination of separate generic methods (Markov Chains, Monte Carlo methods

and MLE) developed over time, how these methods should be combined with

each other to solve the problem at hand is not clear. This makes implementing

their estimation process involved. In this section we detail the method along with

parameters that worked optimally for us. We start by describing each of these

components separately and then combining them to get a reliable initial estimate

of the DDM parameters.

3.3.4.1 Monte Carlo Methods

Despite their simplicity, Monte Carlo methods are extremal powerful numerical

methods that can be applied to a broad set of problems in optimization, integration

and sampling. In our investigation we are specifically interested in computation of

expectations, as most problems can be converted into a computation of expectations.

Let’s say we have a random variable X ∈ RN , with probability density function

(pdf) X ∼ p(x) defined over some set Ω. Let’s say we would like to find the

expectation of a function f(X), which can be computed as:

E[f(X)] = µ =

∫x∈Ω

f(x)p(x)dx (3.17)

34


The Monte Carlo estimate of the expectation can be computed as:

µ =1

n

n∑i=1

f(xi) (3.18)

where, (x1, ..., xn) are n-sample of X ′s. By the weak law of large numbers, if this

expectation exists, then for an arbitrarily small ε:

limn→∞

P (|µ− µ| ≥ ε) = 0 (3.19)

The variance of Monte Carlo estimate is:

V ar(µ) =V ar(f(X))

n(3.20)

i.e., the Monte Carlo method shows 1/sqrt(n) rate of convergence. Even though

the integral involved in the computation of the expectation is a multidimensional

one, the expression of the Monte Carlo estimate is independent of dimensions. If

we attempt to estimate the integral by discretizing the space, the complexity of

the problem will rise exponentially with the number of dimensions. The Monte

Carlo method essentially solves the curse of dimensionality problem. Keep in

mind that the variance in the Monte Carlo estimate is directly proportional to

V ar(f(X). Therefore, in practice, when sampling the X ′s, the nature of the

function f(X);X ∼ p(x) and the number of dimensions play a big role. In the

next section we will discuss how the simple Monte Carlo method can be used to

solve many practical problems.

3.3.4.2 Integration and Optimization using Monte Carlo methods

Lets say we would like to integrate a function q(x) from a to b and the integral

exists but has no analytical solution. The Monte Carlo approximation to the

problem can be obtained by converting the integral to an expectation with respect

to a uniformly distributed random variable U between a and b:∫ b

a

q(x)dx

= (b− a)

∫ b

a

q(x)1

b− adx

= (b− a)

∫ b

a

q(x)pU(x)dx

= (b− a)E(q(U)) ≈ (b− a)

n

n∑1

q(xi);X ∼ U(a, b)

35


Now consider the same function q(x) defined between a to b. We would like to

solve the optimization problem:

x0 = argminxq(x) (3.21)

The Monte Carlo approximate solution can be obtained by uniformly sampling

from U and computing

x∗0 = argminx∈x1,...,xnq(x);X ∼ U(a, b) (3.22)

It should be clear from the formulation of the problems that choosing the right

sampling function, factoring in the shape of the original function q(x) is essential

to keep the variance of the estimates low enough to be of any practical use. The

essential idea to solve the problem efficiently is to spend more time (i.e. give

more density) to the important regions in the sampling space. Many intelligent

ways of sampling exist to minimize the variance of estimation including rejection

sampling, importance sampling and cross entropy method. The problem with

these sampling methods is that they do not scale well with increasing number of

dimensions. The MCMC method essentially addresses these issues by generating

samples while exploring the sampling space using a Markov chain mechanism

which ensures that most time is spent in the important regions in the space. The

methods are detailed in the next sections.

3.3.4.3 Markov chains

To carry out MCMC, the random sequence of Xi from the state space Ω are

generated stochastically in such a way that they follow the Markovian condition:

p(Xi+1|Xi, ..., X1] = p(Xi+1|Xi) (3.23)

Specifically, we are interested in a Markov chain where the conditional distribution

of Xi+1 given Xi is independent of i. The marginal distribution of X1 is called the

initial distribution and the conditional distribution of Xi+1 given Xi is then the

stationary transition probability. If the state space is finite, then the transition

probabilities can be expressed as a matrix K(i, j) with elements pij defined as:

P (Xn+1 = xj|Xn = xi) = pij (3.24)

36


and the initial distribution can be expressed as a vector:

P (X1 = xi) = λi (3.25)

If the state space is uncountably infinite, the initial distribution vector and transition

matrix are replaced by unconditional and conditional probability density functions.

The most important property of the Markov chain is that if the chain satisfies the

conditions for irreducibility and aperiodicity (discussed later), then the chain will

have a stationary distribution π(x) and the transition matrix will converge to this

distribution as limn→∞. Aperiodicity implies that the chain is symmetric in

time, i.e. transition from Xi to Xj and Xj to Xi are equally probable. This is

achieved when:

πipij = πjpji (3.26)

The chain is irreducible, if it is possible to get to any state from any state

irrespective of the current state. This is ensured when the transition matrix

satisfies:

πK = π (3.27)

Therefore by ensuring these conditions, it is possible to generate samples from

the stationary distribution π using the Markov chain. There are many sampling

methods based on the Markov chain with the above mentioned properties. We

used the Metropolis-Hasting sampling to generate samples which is discussed in

the next section.

3.3.4.4 Sampling using Metropolis-Hasting update

Lets say we would like to sample from a distribution p(x), such that the density

is known only upto a scaling factor. In other words, p(x) is non-negative and

integrates to a finite non-zero value. The Metropolis-Hasting (Metropolis et al.,

1953; Hastings, 1970) update works as follows:

• Given the current state x, propose a transition to y, having a proposal

conditional probability given x donated by q(y;x)

• Accept the transition with probability

min(1,p(y)q(x; y)

p(x)q(y;x))

37


• Else next state is a copy of the current state.

if q(y;x) > 0 with probability 1, then this method ensures that the generated

chain satisfies the necessary conditions to obtain a stationary distribution p(x).

The nature of the conditional density q(y;x) decides how fast the process diffuses

through the sampling space. The normal distribution is quite often used as a

generic proposal density q(y;x) = N(x, σ2). In this case a large σ results in many

rejections while a small σ results in slow diffusion. In the next section we will

discuss how these methods can be combined to get a good estimate of parameters

of the DDM model from RT data.

3.3.4.5 MCMC - MLE applied to DDM

Given a set of RT observations x, the likelihood that a DDM parameterized by Θ

generated it is:

L(Θ|x) = P (X = x|Θ) (3.28)

The MLE tries to find the Θmle that maximizes the likelihood of the observed

data, i.e.

Θmle = argmaxΘL(Θ|x) (3.29)

In practice it is convenient to work with the log of the likelihood function. The log

function being a monotonically increasing function does not affect the analysis.

So the optimization function is changed to:

Θmle = argmaxΘln(L(Θ|x)) (3.30)

The advantage of MLE is that it shows the following important properties:

• Consistency: It converges almost surely to the true value of Θ as the sample

size tends to infinity.

• It achieves the CRLB when sample size tends to infinity.

These properties make the MLE an ideal candidate for parameter estimation.

If we had access to closed form solution to the MLE expression we would have

computed it directly without relying on simulations. For now we will use the

MCMC to generate an estimate of the MLE. As the actual value of the likelihood

is of little importance, we can construct a likelihood ratio function as:

Λ(Θ0) =L(Θ0|x)

L(Θ1|x)(3.31)

38


the ratio captures how many time more likely the data x is coming from Θ0

compared to Θ1. As we do not have access to a closed form expression for

likelihood we have to use a proxy function. We can use the χ2 measure as described

earlier in section 3.3.3 to indirectly measure the likelihood. Formally:

P (Θ|x) ∝ exp(−χ2Θ) (3.32)

Therefore the likelihood ratio changes to

l(Θ) = exp(χ2Θ1− χ2

Θ0) (3.33)

We now construct a Markov chain on the parameter space Θ using the Metropolis-

Hasting update. We take the stationary distribution of the chain to be p(Θ) =

P (Θ|x) ∝ exp(−χ2Θ). We can generate samples of Θ from the distribution and

simply compute the Monte Carlo approximate estimate of the MLE. Note that χ2

measure is computed via simulations as described earlier. Each simulation requires

approximately 104 simulated RTs. If we use the previously described simulation

method for generating RTs, each RT will require approximately 500 cycles. To get

reliable estimates from MCMC-MLE we observed that approximately 500 cycles

are required in the Markov chain and another 500 cycles are required to actually

estimate the parameters using previously described method. In other words,

estimation of parameters based on previous literature require approximately 109

operations. Another practical problem using this method is that the data is

usually censored due to limitation of time. In case of PVTs the censor time is

usually 10 seconds. A much faster and elegant solution to the problem is to

directly obtain a closed form solution to the DDM. In the next section we will

derive a closed form solution for the DDM and propose a MLE based on it to

quickly and efficiently estimate the model parameters from data.

3.4 Improvements in Estimation

3.4.1 DDM: Closed form solution

In this section we derive a closed form solution for the model. The complete

distribution for the model is a compound probability distribution which can be

derived from Equation 3.11. The resulting distributions are defective (i.e. they

have finite mass at infinity), but nonetheless can be used to get a maximum

39


likelihood estimate of the parameters. Starting from standard diffusion process

with drift µ and volatility σ2 originating at 0 with process termination boundary at

a, the distribution of RT is f(t) as derived earlier. Now assuming µ ∼ N (ξ, η) , the

resulting compound distribution is given by:

g (t|a, ξ, η) =

∫ ∞−∞

f(t|µ)p (µ) dµ, t > 0 (3.34)

g (t|a, ξ, η) =

∫ ∞−∞

a

σ√

2πt3e

− (a−µt)2

2σ2t

(1√2πη2

)e− (µ−ξ)2

2η2 dµ, t > 0 (3.35)

The integration can be done by manipulating the quadratic forms in the

exponents to yield:

g (t|a, ξ, η) =a√

2πt3(η2t+ σ2)e

− (a−ξt)2

2t(η2t+σ2)

, t > 0 (3.36)

g (t|a, ξ, η) can be taken to be 0 for t ≤ 0. Then the cumulative distribution

function G(t|ξ, η) is given by:

G (t | a, ξ, η) =

1− Φ

(a−ξt√t2η2+σ2t

)+ e

2aξ+2a2η2

σ2 Φ

(−aσ2+2atη2+σ2ξt

σ2√t2η2+σ2t

), t > 0

0, t ≤ 0(3.37)

Where Φ is the cumulative distribution function of the standard normal distribution.

In the DDM, the distribution Equation 3.36 is shifted because of the non-decision

time. Let the shift in distribution be t0. Then Equation 3.36 can be appropriately

modified to:

g (t|a, ξ, η, t0) =

a√2πt3k(η2tk+σ2)

e

− (a−ξtk)2

2t(η2tk+σ2)

, tk=t−t0, tk>0

0, tk ≤ 0

(3.38)

As per the DDM, t0 ∼ U(Ter − St

2, Ter + St

2

), the resulting compound distribution

will be:

r (t | Θ) =

∫ Ter+St2

Ter−St2

g(t|a, ξ, η, t0)p (t0) dt0 (3.39)

40


r (t | Θ) =

(1

St

)∫ t−Ter+St2

t−Ter−St2

g (tk | a, ξ, η) dtk (3.40)

r (t | Θ) =

(1

St

)(∫ t−T er+St2

−∞g (tk | a, ξ, η) dtk −

∫ t−Ter−St2

−∞g (tk | a, ξ, η) dtk

)(3.41)

r (t | Θ) =

(1

St

)G

(t− T er +

St2|a, ξ, η

)−G

(t− T er −

St2|a, ξ, η

)(3.42)

Equation 3.42 captures the defective probability density function for the one-

choice DDM. Instead of relying on simulation, the model parameter (Θ) can now

be estimated using an MLE. Given a set of observation t1, t2, . . . , tn the log

likelihood of parameter vector Θ can be computed as:

lnL (Θ | t1, . . . , tn) =n∑i=1

ln (r (ti | Θ)) (3.43)

If the experiment is censored at t = tc, and there are m censored observations

along with n uncensored observations the log likelihood will be modified to:

lnL (Θ | t1, . . . , tn, tc1, . . . , tcm) =n∑i=1

ln (r (ti | Θ)) + mln (1−R (tci | Θ)) (3.44)

the minimum negative log likelihood is estimated using interior-point algorithm

(Dantzig, 1965). R (tci | Θ) is computed using trapezoidal numerical integration.

For starting the estimation process the boundary a is chosen randomly. Initial

value of mean drift ξ0 and mean non-decision time T 0er is computed as:

ξ0 = 3

√aσ2

σ2RT

; T 0er = µRT −

a

ξ0(3.45)

here µRT is the mean and σ2RT is the variance of observed RTs. These expressions

are obtained by assuming a specific condition in the model where η = 0, St = 0

(Equation 3.38). The starting values of η and St can be chosen as a suitable

fraction of ξ0 and T 0er respectively based on domain knowledge. We refer to the

proposed estimator as MLE based estimator. In the next section we compare the

proposed simulation and estimation methods with previously available ones.

41


3.5 Comparison

3.5.1 Methods

The proposed mixed simulation method which is a combination of inverse Gaussian

and random walk simulation is compared with previously proposed method by

Ratcliff & Van Dongen (2011) relying solely on random walk approximation (from

here on referred to as random walk simulation). Since, the modified simulation

proposed partly relies on exact sampling method it is guaranteed to be more

accurate than previously proposed method based on random walk approximation

only. The estimators were compared both in terms of speed and efficiency. Both

estimators were checked for any bias in estimation. To carry out the analysis we

used six standard parameter sets as shown in Table 3.1.

Set a Ter(ms) St(ms) ξ η1

0.09

150 30 1.1 0.202 170 40 1.0 0.253 190 50 0.9 0.304 210 60 0.8 0.355 230 70 0.7 0.406 250 80 0.6 0.45

Table 3.1: Standard parameter sets

For simulation 20,000 RT samples were generated for each set using both

simulation methods. For the estimation analysis we chose all possible combinations

of drift and variability in drift (ξ and η) for each set from the table. This allowed

36 combinations of ξ and η for each set in Table 3.1, i.e. a total of 216 estimations.

The values for Table 3.1 were arrived at based on observed estimated parameter

values for subjects on a PVT before and after sleep deprivation (discussed in later

chapters). A 10 minute PVT typically results in 95 responses. Set 6 represents

a poorly performing subject while set 1 represents a very fast subject. To make

the estimation process realistic, the RTs were generated by simulating a PVT.

Three sets of estimation were carried to represent 10 minute, 20 minute and 30

minute PVT session. The analysis was carried out on a run on a contemporary

workstation with 6 core Intel Xeon 3.2 GHz processor with 16 GB of RAM.

The algorithms were implemented in Matlab R2013b and were vectorized for

performance.

42


3.5.2 Comparison of simulation methods

For the simulation, the step size for random walk approximation was fixed at

τ = 0.5ms. The simulations were censored at t=10 sec. Figure 3.3 shows the

comparison between the two simulators for the 6 sets.

0.42

0.4

0.57

0.83

2.1

4.3

0

20

40

60

80

100

120

140

1 2 3 4 5 6

Ra

tio

of

sim

ula

tio

n s

pe

ed

s

Set

Simulation Speed Comparision

Figure 3.3: Ratio of simulation speeds of random walk simulation to mixedsimulation method. The numbers shown are actual simulation speed of theproposed mixed simulation method in seconds. As we move from set 1 to 6,the mean drift goes on reducing. As a result the proportion of sampled drift thatis less than zero also increases. This in-turn reduces the speed of mixed simulation.Under normal circumstances, mixed simulation is two order of magnitudes fasterthan random walk simulation.

3.5.3 Comparison of estimation methods

For carrying out the χ2 based estimator we used both the random walk based

simulation as well as the proposed mixed simulation method. The estimation

speed for all three estimation methods is shown in Table 3.2. The estimation

speed for the original χ2 based estimator with random walk simulation was too

slow; therefore, it is reported only for a 10 minute PVT set. Table 3.3 summarizes

the mean and standard deviation of error in estimation, where error in estimation

e was defined as e = Θ− Θ. The estimates were unbiased based on a one sample

t-test at significance level α = 0.05. Normalized DDM parameters are shown.

The efficiency of both the estimators is similar, although the proposed simulator

is an order of magnitude faster and much simpler to implement.

43


SetTotal estimation time (for 216 estimations)χ2 based estimator χ2 based estimator

MLE based estimator(random walk) (mixed simulation)

10 Min. PVT ≈ 90 hours 12 Hours 12 Minute20 Min. PVT N.A. 12 Hours 11 Minute30 Min. PVT N.A. 10 Hours 10 Minute

Table 3.2: Estimation speed. The χ2 based estimator using random walksimulation is the original method as proposed by Ratcliff et al., 2011. As theestimation speed was prohibitively slow, we used the proposed mixed simulationmethod with the χ2 based estimator which while still slow, significantly improvedspeed. The proposed MLE based estimator does not require any model simulation.

Set Parameter χ2 based estimator MLE based estimator

10 Min. PVT

Ter(ms) −1.17± 10.8 −0.8± 15St (ms) −1.2± 18.8 0.9± 15.5ξ/a −0.10± 1.42 −0.03± 1.57η/a −0.06± 0.98 −0.03± 1.03

20 Min. PVT

Ter(ms) −0.94± 9.8 1.4± 14St (ms) −1.2± 18 0.9± 13.4ξ/a −0.09± 1.13 −0.01± 1.38η/a −0.07± 0.94 −0.09± 0.72

30 Min. PVT

Ter(ms) −0.25± 8.8 0.33± 13.5St (ms) 1.4± 18.5 1.0± 11.9ξ/a −0.08± 1.00 0.00± 1.00η/a −0.07± 0.08 0.03± 0.67

Table 3.3: Mean and standard deviation of error in estimation. Drift parametersare normalized.

3.6 Cramer-Rao Lower Bound

The CRLB (Radhakrishna Rao, 1945) measures the lower bound on the variance

of estimator. Suppose Θ has to be estimated from the available data x, which

is generated according to an underlying distribution p(x; Θ). Let the estimator

denoted by g(X) be an unbiased estimator of p(x). As per the Rao bound, the

variance of an unbiased estimator E(g(X)) = Θ of Θ is bounded by the reciprocal

of Fisher information I(Θ). That is:

var(Θ) ≥ I(Θ)−1 (3.46)

where the Fisher information is computed as:

I(Θ) = E

[(∂ln(L(x; Θ))

∂Θ

)2]

= −E

[∂2ln(L(x; Θ))

∂Θ2

](3.47)

44


where L is the likelihood function defined earlier in Equation 3.28. The equations

are valid subjected to two regularity conditions:

(i) For all x such that p(x; Θ) > 0,

∂

∂Θlog p(x; Θ)

exists and is finite.

(ii) The integration operation with respect to x and differentiation with respect

to Θ can be interchanged in the expectation of the estimator.

Here, Θ is assumed to be a scalar. In case it is a vector in Rn, then the variance

in estimation will be replaced by a covariance matrix in Rn×n. The bounds on the

covariance is:

covθ (g(X)) ≥ ∂Eg((X))

∂θ[I (θ)]−1

(∂Eg((X))

∂θ

)T(3.48)

Where I is the Fisher information matrix in Rn×n with element Ii,j defined as

Ii,j = E

[∂

∂θilog p (x;θ)

∂

∂θjlog p (x;θ)

]= −E

[∂2

∂θi∂θjlog p (x;θ)

]. (3.49)

In spite of the closed form solution for the likelihood function as derived in

Equation 3.43, the derivation of the CRLB is highly involved due to the complex

form of the equation. Even if we assume that the covariance structure of estimate

has only diagonal elements (i.e. we can compute CRLB with respect to individual

parameters assuming other parameters are held constant), the computation of

the CRLB is still difficult. Fortunately from the perspective of our investigation,

we are not interested in the closed form expression for the CRLB. Instead, much

insight can be gained about the practical limitation of the estimator using the

maximum likelihood estimator. The MLE asymptotically approaches the CRLB

as the available data approaches infinity. Therefore, we design some simulations

that would give practical insights into capabilities of the DDM in realistic scenarios.

45


3.6.1 CRLB estimates using simulations

In this section we are interested in the limitations of the estimator in realistic

scenarios. Specifically, we are interested in finding out an approximate estimate of

the CRLB and how low the error in the estimates go as we incrementally increase

the data size. Getting one set of PVT data is the most ideal, while carrying

out more than four PVTs is too cumbersome and unrealistic. Therefore, we

compute the errors in MLE estimate as we gather up-to four sessions of standard

10-minute PVTs. To obtain an estimate of the error variance and bias, each PVT

was simulated 100 times and the parameters were estimated for each. The error

bias and variance can then be computed as:

biasΘ = Θµ =

∑100i=1(Θi − Θi)

100varΘ =

∑100i=1(Θi −Θµ)2

99(3.50)

We also obtain estimates with 10,000 simulated RTs. CRLB was estimated using

numeric differentiation of the partial derivatives of log likelihood (Equation 3.47)

and monte carlo method to obtain the expectation (Equation 3.18). CRLB was

computed for each parameter independently (i.e. covariance structure of error

is assumed to have only diagonal elements). All estimates were obtained for

a fixed parameter set: a = 0.01, ξ = 1.0, η = 0.25, Ter = 160ms, st = 50ms.

This parameter set represents an average baseline measurement. Notwithstanding

the fact that CRLB as well as the performance of the estimator will depend on

the exact value of the parameters, this analysis gives an idea of the expected

error in estimation without overcomplicating the study with rigorous analysis.

Furthermore, the performance of the estimator is essential to gauge the sample

size for a given clinical study.

3.6.2 Error estimates: Results

Figure 3.4 shows the standard deviation of error estimates along with the CRLB

estimates. Except for range of non-decision time st, the error in estimates remain

comparatively higher than the lower bound even when data size was increased up-

to 10,000 RT samples. The error in the estimates are also important to find out

the sample size for a clinical study. For instance, if we want to compare the mean

DDM parameters between two groups. Let us assume the following requirements

for the study:

46


• significance level α = 0.05.

• power of the experiment 1− β = 0.8.

• lets say we would like to be able to measure a difference in mean normalized

diffusion drift of 1.

• the standard deviation in mean normalized drift within each group is 1.

The sample size n can be computed as:

n = 2 ∗ d(Zα/2 + Zβ)22s2

(∆µ)2e (3.51)

where d.e is the ceiling function, Z is the Z-statistic and s2 is the variance of

measurement, which in our case will be the sum of variance within the group

(assumed 1 here) plus the variance due to error in estimation (which will depend on

how many PVTs we collect). The equation is derived using central limit theorem

(i.e. the sampling distribution of mean is distributed normal with mean equal to

population mean and standard deviation σsample = σpopulation/√

(n). substituting

the appropriate value into the equation we get a value of n=166 using 1 session

of PVT, n=140 using 2 sessions of PVT and n=92 using 3 sessions of PVT.

3.7 DDM: Limitations

Despite the many advantages of the DDM, there are limitations of the model

that must be kept in mind when applied to different studies. There are three key

limitations of the DDM:

(i) All parameters of the model cannot be uniquely identified (Ratcliff & Van Dongen,

2011). From a statistical point of view, it can be assumed that the model has

four parameters instead of five and it will not make any difference from the

point of view of estimation or simulation. But it is important to keep in mind

that each parameter has an underlying neuronal architecture that mediates

the decision process. Therefore, a four parameter normalized DDM is not

equivalent to a five parameter DDM from a clinical and neuro-biological

perspective.

47


(ii) The DDM is based on accumulator models which makes the assumption that

distinct components of decision process are strictly serial (H. R. Heekeren et

al., 2008). This is known to be false from previous experiments. Although,

in most situations the assumption has little effect in terms of model fit

(Ratcliff, 1978).

(iii) The model assumes that each RT is IID according to the underlying DDM.

Despite the good model fit between measurements and the model it is known

that RTs are not independent of each other. The RTs are strongly affected

by ISI as well as time on task (H. Van Dongen et al., 2011).

In our investigation, the IID assumption violations could have the most impact.

Further improvements in the DDM can be made to accommodate the additional

parameters. Currently, there is very little clinical and neurophysiological results

available to guide the changes required in the model. In the absence of a model

0

0.2

0.4

0.6

0.8

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1.2

1.4

PVT 10 min PVT 20 min PVT 30 min PVT 40 min 10,000

samples

err

or

sta

nd

ard

de

via

tio

n

0

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1.4


samples

err

or

sta

nd

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via

tio

n

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samples

err

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nd

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tio

n (

ms)

0

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18

PVT 10 min PVT 20 min PVT 30 min PVT 40 min 10,000 samples

err

or

sta

nd

ard

de

via

tio

n (

ms)

Estimates of normalized drift (ξ/a) Estimates of normalized across trial variability in drift (η/a)

Estimates of mean non-decision time (Ter) Estimates of range of non-decision time (St)

CRLB

Estimates

Figure 3.4: Standard deviation of error as progressively more data is used forestimation. The CRLB estimate shows the lowest possible error achievable withthe given amount of data. The analysis is done for a fixed value of parametersthat represent an average subject under baseline condition.

48


that takes into account the temporal and spatial structure of the RTs, we will

consider alternate ways of looking at the RT distributions along with the DDM

in the next chapters.

3.8 Conclusions

The one choice DDM is a powerful model of perceptual decision making. The

availability of simple simulation and estimation methods is essential for the current

investigation as well as the clinical community to fully leverage the model for

real world applications. We brought to notice the simple observation that the

simulation time can be considerably improved by switching between sampling from

inverse-Gaussian distribution for positive drifts and falling back on random walk

simulation for negative or zero drifts. The novelty of this chapter lies in deriving

a closed form solution for the model which did not exist earlier in literature. The

closed form solution for the model help gain insights into one choice perceptual

decision making and help design more efficient and fast estimators. We used the

maximum likelihood estimator, which is a well-established method for parameter

estimation to show that the estimated parameters based on the close form solution

are in close agreement with previous method based on data fitting. The estimation

speed of the MLE based estimator is at-least an order of magnitude faster while

maintaining the same level of efficiency compared to previously proposed method.

We also derived error estimates and CRLB on estimates which will be very helpful

in comparing and setting an upper limit for future improvements in the DDM

estimator. The error estimates are also very important in designing clinical

experiments. In the next chapter we will use the methods developed here to

answer some of the key questions with respect to the DDM applied to study

vulnerability to sleep deprivation.

49

Chapter 4

DDM Parameters andDifferential Vulnerability to SleepDeprivation

4.1 Introduction

In the previous chapter we discussed efficient ways to simulate and estimate the

model parameters for one-choice DDM. As discussed earlier, an attractive feature

of DDM is that it can predict the response time distribution under different

contexts (Ratcliff, 2002) and varying levels of noise (Ratcliff & Tuerlinckx, 2002).

Among the numerous studies using the PVT to characterize performance in sleep-

deprived persons, there exists only one (Ratcliff & Van Dongen, 2011) that used

diffusion modeling to explain behavior. In this chapter, we ascertained if the

diffusion model could differentiate persons according to their vulnerability to SD

as measured by a decline in psychomotor vigilance. To fill these gaps in our

understanding of behavior following SD we posed two questions:

(i) Are the parameters of the diffusion model differentially affected in vulnerable

and resistant participants? and

(ii) Can diffusion parameters determined prior to sleep deprivation predict performance

following SD?

The analysis presented in this chapter appears in Patanaik, A., Zagorodnov, V., Kwoh,C. K. and Chee, M. W. L. “Predicting vulnerability to sleep deprivation using diffusion modelparameters.”, Journal of Sleep Research, 2014. doi: 10.1111/jsr.12166. Used with permission.

50

Chapter 4. DDM Parameters and Differential Vulnerability to SleepDeprivation

4.2 Methods

4.2.1 Subjects

A total of 135 participants (69 females, mean age 21.9±1.7 years) from five

different functional imaging studies (Chee & Chuah, 2007; Venkatraman et al.,

2007; Chee et al., 2010; L. Y. Chuah & Chee, 2008; L. Y. Chuah et al., 2010)

on sleep deprivation contributed behavioral data to this chapter. The 5 studies

shared common recruitment criteria and protocol for sleep deprivation. Volunteers

had to:

(i) be right handed,

(ii) be between 18 to 35 years of age,

(iii) have habitual good sleeping habits (6.5 to 9 hours of sleep every day),

(iv) have no history of sleep or psychiatric or neurological disorders and

(v) have no history of severe medical illness.

All participants indicated that they did not smoke, consume any medications,

stimulants, caffeine or alcohol for at-least 24h prior to the sessions. Informed

consent was obtained from all participants in accordance to study protocols approved

by the National University of Singapore Institutional Review Board.

Participants visited the laboratory 3 times. They first attended a briefing

session during which they were informed of the study protocol and requirements

and were practiced on the study task. At the end of this session, each participant

was given a wrist actigraph to wear throughout the study. The first experimental

session took place approximately a week later. The order of the 2 experimental

sessions (rested wakefulness and sleep deprivation) was counterbalanced across all

the participants and separated by 1 week. This was to minimize the possibility of

residual effects of sleep deprivation on cognition for those participants whose sleep-

deprivation session had preceded their rested-wakefulness session (H. P. Van Dongen

et al., 2003). Sleep duration was verified by actigraphic data and data from non-

compliant subjects were not analyzed.

51


4.2.2 Experimental Details

In the rested wakefulness (RW) session, subjects arrived at the laboratory on

scheduled date at 07:30. The PVT was administered at 08:00. For sleep deprivation

sessions, subjects arrived at the laboratory on scheduled date at 19:30. They

underwent a night of SD under supervision of a research assistant. PVT was

administered every hour from 20:00 till 5:00 next morning (10 test periods). For

this report only data from the first two periods taken at 20:00 and 21:00 during

the wake maintenance zone of the SD session and two test periods at 4:00 and 5:00

following a night of total sleep deprivation were analyzed. We did not compare

RW with SD directly because only one data point was available, as discussed in

the previous chapter this was too small for our analysis. The first two sessions

were labeled Evening before Sleep Deprivation (ESD) and the last two sessions

were labeled sleep deprivation (SD). Subjects also rated their subjective sleepiness

on the 9-point Karolinska Sleepiness Scale after each PVT test. Through the

night, subjects were allowed to engage in non-strenuous activities such as reading,

watching videos and conversing. Subjects were instructed to respond as quickly

and as accurately as possible. RTs less than 150ms were regarded as false alarms,

and they were excluded from analysis. All PVTs administered were of 10-min

duration.

4.2.3 DDM parameter Estimation

The model parameters were estimated using a mixed estimation process as detailed

in the previous chapter. The MLE based estimator was used to get an initial

estimate of the parameters which were fine tuned using the χ2 based estimator.

As discussed earlier, while estimating the parameters only drift ratios (ξ/a, η/a)

are uniquely identifiable. For the sake of simplicity, from this point onwards, in

this chapter, we will use drift and drift ratio interchangeably. When comparing

the drift parameters across group, it is implicit that they were normalized by the

estimated boundary parameter.

4.2.4 Statistical analyses

Group differences were evaluated using an independent samples t-test. To assess

the interaction effect of state (ESD or SD) and group (vulnerable or resistant) on

52


relevant diffusion parameters, a 2×2 factorial design analysis of variance (ANOVA)

was employed. Alpha was set at 0.05. To assess the discriminative power of

the diffusion parameters a binary logistic regression analysis was performed to

classify subjects into vulnerable and resistant groups using baseline data. We used

vulnerability as a dependent variable and diffusion model parameters measured

in ESD as independent variables. We also tried introducing baseline standard RT

metrics to the set of independent variables, anticipating any increase in accuracy.

In the set of standard RT metrics, we also included the slowest 10% response speed

(RS is reciprocal of reaction time). The slowest 10% RS is known to be correlated

strongly with drift parameter (Ratcliff, 2002). The independent variables were

introduced one at a time sequentially, using the forward selection method, until

the addition of an extra variable resulted in no statistically significant increase

in accuracy. A receiver operating characteristic (ROC) curve was obtained by

varying the threshold of the logistic function. All analyses were conducted using

SPSS version 20 (IBM, Chicago, IL, USA) and Matlab 2013b (The MathWorks,

Inc., Natick, MA, USA).

4.3 Results

4.3.1 Identification of vulnerable and resistant subjects

Based on the change in the number of lapses, l, between SD and ESD (δl =

lSD − lESD), subjects were identified as resistant if they belonged to the lower

tertile, and as vulnerable subjects if they belonged to the upper tertile. A lapse

was defined as a trial with RT ¿ 500ms. Resistant subjects (n=43) had δl < 4,

and vulnerable subjects (n=45) had δl > 12.

The two groups were similar in age (mean age for resistant group = 22.0

years, std dev = 1.97 years; mean age for vulnerable group = 22.0 years, std

dev. = 1.68 yrs; t86 = 0.05, n.s.), and gender (18 females in resistant group,

22 females in vulnerable group; χ21 = 0.43, n.s.). As expected, across the entire

group, sleep deprivation (SD) elicited significant changes in mean and median RT,

the reciprocal of RT and lapses compared to the evening before sleep deprivation

(ESD) state (Table 4.1). SD had a significant effect on diffusion model parameters

(drift, non-decision time) other than within trial variability in drift. Importantly,

during ESD, there was no significant difference in any of the traditional RT metrics

between the two groups (smallest p=0.08, for total lapses; Figure 4.1).

53


Table 4.1: Standard reaction time and diffusion parameter statistics of studyparticipants. All standard metrics were significantly affected by sleep deprivation.In terms of diffusion parameters, except for standard deviation in diffusion drift,all other parameters were significantly affected.

ESD(n=135) SD(n=135) P-valueReaction time (RT) statistics

Mean RT, ms 266 ± 34 442 ± 432 <0.001Mean response speed, 1/sec 4.0± 0.4 3.3± 0.6 <0.001Median RT, ms 249 ± 26 312 ± 138 <0.001Total lapses 2.6 ± 3.7 15.8 ± 17.9 <0.001

Diffusion parameter statisticsDrift ξ 0.983 ± 0.246 0.697 ± 0.242 <0.001Across trial standard deviation in drift η 0.249 ± 0.153 0.222 ± 0.140 0.13Mean non-decision time Ter, ms 157 ± 17 165 ± 22 <0.001Range of non-decision time St, ms 47 ± 16 56 ± 22 <0.001

Overall χ2 fit for evening before sleep deprivation (ESD) state was 16.8 ± 7.2and sleep deprivation (SD) state was 17.7 ± 9.7. Critical value for χ2, df = 14 is

26.1.

4.3.2 Effects of state and group on diffusion parameters

Mean diffusion drift: There was a significant main effect of state on mean diffusion

drift ξ (F1,172 = 69.4; p<0.001). Averaged across the two tertiles, diffusion drift

was significantly lower in SD state compared to ESD state. Group had a significant

main effect on drift, with vulnerable participants having a lower mean diffusion

drift ξ than resistant participants, F1,172 = 45.1; p<0.001. Vulnerable subjects

had significantly lower diffusion drift in both ESD (t86 = 2.44; p<0.05) and SD

(t86 = 7.65, p<0.001) compared to resistant subjects.

The interaction of state and group was significant, (F1,172 = 8.33 p<0.005)

(Figure 4.2). Thus, while both groups showed significant declines in mean diffusion

drift following sleep deprivation (vulnerable t88 = 9.02, p<0.001; resistant: t84 =

3.3, p<0.005), decline in drift rate was greater in vulnerable than in resistant

subjects.

Mean non-decision time: There was a main effect of state on mean non-

decision time (Ter) (F1,172 = 7.47; p<0.01); with Ter being faster in ESD than

during SD. This was modulated by group (group x state interaction (F1,172 =

6.1 ; p<0.05)), such that the decline in non-decision times was also significant

in vulnerable subjects (t88 = -3.54; p<0.001, but not in resistant subjects (t84 =

0.11, p=0.91). There was no main effect of group on Ter, F1,172 = 1.45; p=0.23.

54


200

210

220

230

240

250

260

270

280

290

res vul

me

an

RT

, (m

s)

200

210

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res vul

me

dia

n R

T,

(ms)

0

0.5

1

1.5

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2.5

3

3.5

4

4.5

res vul

tota

l la

pse

s

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3.9

3.95

4

4.05

4.1

4.15

res vul

me

an

re

spo

nse

sp

ee

d,

(1/s

ec)

Figure 4.1: Performance metrics measured on the evening before sleepdeprivation (ESD). Metrics include mean reaction time (RT), mean responsespeed (RS=1/RT), total lapses and median RT for subjects vulnerable (vul)and resistant (res) to sleep deprivation. Although the resistant group performedbetter than the vulnerable group in the ESD state, none of the RT metrics werestatistically significantly different between the two groups (smallest p=0.08 formean RT). Error bars represent one standard error of the mean.

4.3.3 Predicting vulnerability from baseline data

We observed a classification accuracy of 69.3% at a sensitivity of 65.1% and

specificity of 73.3% using baseline normalized diffusion drift (ξ/a), diffusion signal

to noise ratio (ξ/η) and mean non-decision time (Ter). The baseline slowest 10%

RS was observed to be highly correlated (r = 0.77, p<<0.001) with baseline

mean diffusion drift. Despite this, no improvement in classification was observed

by addition of any traditional RT measures. The ROC curve is presented in

Figure 4.3. The area under the curve was 0.74.

55


5

6

7

8

9

10

11

12

13

ESD SD

me

an

dif

fusi

on

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ft (

no

rma

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d),

ξ/

a

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res

vul

1.5

1.7

1.9

2.1

2.3

2.5

2.7

2.9

3.1

3.3

ESD SD

acr

oss

tri

al

dri

ft v

ari

ab

ilit

y (

no

rma

lize

d),

ƞ/a

State

150

155

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ESD SD

no

n-d

eci

sio

n t

ime

, T

er

(ms.

)

State

40

45

50

55

60

65

ESD SD

ran

ge

of

no

n-d

eci

sio

n t

ime

, S

t(m

s.)

State

Figure 4.2: Mean estimated diffusion parameters. Estimated for vulnerable (vul)and resistant (res) groups in the evening before sleep deprivation (ESD) andafter sleep deprivation (SD). In ESD, vulnerable subjects showed significantlylower (p<0.05) mean diffusion drift compared to resistant subjects. In addition,following SD, vulnerable subjects also displayed significantly lower drift (p<0.001)and significantly higher non-decision time (p<0.01). Both mean diffusion drift andmean non-decision time showed statistically significant group by state interaction(p<0.005 and p<0.05 respectively). Error bars represent one standard error ofthe mean.

4.4 Discussion

4.4.1 Effect of sleep deprivation on diffusion parameters

Our results replicate and extend the previous finding that single boundary diffusion

drift ratio is reduced with sleep deprivation on the standard Psychomotor Vigilance

Task. Diffusion drift has also been found to be reduced on a numerosity discrimination

task (Ratcliff & Van Dongen, 2009). However, these sleep deprivation related

changes in diffusion drift may be context and/or task-dependent. Menz et al.

(2012) for example, observed that drift reduced with a decision task in SD for

easy but not for hard decisions. Additionally SD appears to affect non-decision

diffusion parameters (mean and variance of non-decision times). The latter finding

is possibly a result of the greater power of the present study (n=135 vs. n=19

for one of the original studies) and is consistent with the notion mooted by those

56


TP

Rate

(S

ensitiv

ity)

FP Rate (1-Specificity)

0 0.2 0.4 0.6 0.8 1

0

.2

0

.4

0.

6

0.8

1

Figure 4.3: Receiver operating characteristic curve obtained by varying thresholdof a logistic regression based classifier. Diffusion parameters estimated on theevening before sleep deprivation (ESD) state were used to predict the vulnerabilityof the subjects to sleep deprivation. The best operating point was found at a truepositive (TP) rate of 65% and false positive (FP) rate of 30.7%. The area underthe curve was 0.74.

authors that SD can affect multiple cognitive processes (Ratcliff & Van Dongen,

2009).

4.4.2 Difference in diffusion parameters between vulnerableand resistant subjects

The most interesting finding of the present work is that the drift parameter in

the evening before sleep deprivation predicted greater vulnerability to SD that

was not anticipated by merely observing PVT performance. The latter may have

arisen because in the ESD state, diffusion drift rate and non-decision time were

57


vul

res

Diffusion drift (normalized)Non-decision time (ms)

Page 1 of 1

5/31/2014file:///C:/Users/Amiya/Desktop/simulate.svg

Figure 4.4: Expected median reaction time (RT) for different combinationsof mean normalized diffusion drift and mean non-decision time generated bysimulating 30 minutes of psychomotor vigilance task (PVT). The variabilityparameters were fixed at the group average values (η/a=2.62,st=52ms). Responsetimes (as measured by median RT) of two representative subjects, one vulnerable(vul) and the other resistant (res) to SD were overlaid. Both subjects had thesame median RT (=256ms) in the baseline Evening Before Sleep Deprivation(ESD) condition.

affected in opposite directions- vulnerable subjects had longer decision times but

shorter non-decision times.

In the ESD state, when the diffusion drift was high, the mean drift and non-

decision parameters traded-off without affecting overall observed performance.

The effects of varying levels of mean diffusion drift and mean non-decision time on

median RT can be modeled (Figure 4.4). The figure was generated by simulating

30 minute PVTs for each pair of diffusion-drift and non-decision time parameters

while holding constant variability parameters. The modeled RT of two representative

58


subjects, one vulnerable and one resistant to SD are shown overlaid on the plot.

Both subjects have different mean diffusion drift and non-decision parameters but

the same median RT in the ESD state. Modeling showed that as the diffusion

drift was reduced, median RT became dominated by the drift parameter and the

tradeoff between the parameters became less apparent.

4.4.3 Interaction effect of state and group on diffusionparameters

Both mean diffusion drift and mean non-decision time showed statistically significant

group by state interaction. The results suggest that in terms of decision time (i.e.

drift parameters) subjects with better performance in ESD state were less affected

after sleep deprivation. Furthermore, those with faster decision time performance

but slower non-decision times in the ESD state continued to be less affected by

sleep deprivation.

The observation that resistant subjects were less affected by sleep deprivation

on all diffusion parameters suggests that they may have greater cognitive reserve

(Stern, 2002) compared to vulnerable subjects (Bell-McGinty et al., 2004; Y. L. Chuah

et al., 2006). This concept has been primarily applied to cognitive aging but has

also been shown to be relevant in the context of sleep deprivation (Mu et al., 2005;

Chee et al., 2006).

4.4.4 Possible neurocognitive accompaniments of reduceddiffusion drift

Although a slower drift rate speaks to slower accumulation of evidence on which

to base action, the present experiments and analyses were not designed to examine

possible contribution of this mechanism. The experiment that most closely examines

rate of processing limitations in the sleep deprived state is one where sleep deprivation

was accompanied by a leftward shift in the frequency response profile of fMRI

signal in higher visual cortex (Kong et al., 2014). This finding indicates that the

maximal rate at which pictures are processed in higher (but not primary) visual

cortex is lowered.

Sleep deprivation has been consistently shown to result in reduced recruitment

of parieto-frontal and visual extrastriate brain regions during the performance of

visual attention tasks (Chee et al., 2010; Lim & Dinges, 2010; Tomasi et al.,

59


2009) and even in the preparatory phase (Chee et al., 2011). The degree of

reduction of activation generally corresponds to impaired accuracy in tasks but

only in two studies has greater baseline activation to mark persons relatively more

resistant to the negative effects of sleep deprivation (Mu et al., 2005; Y. L. Chuah

et al., 2006), in accordance with the cognitive reserve hypothesis. As such, the

neural correlates of predictors of vulnerability to sleep deprivation remain to be

further characterized. Neuro-imaging studies conducted in the resting state could

potentially give more insights. We will address this in a later chapter.

4.4.5 Predictive value of diffusion model parameters

Even though there were statistically significant differences between the vulnerable

and resistant group in the baseline ESD condition, it remains to be seen if the

diffusion parameters are useful in predicting vulnerability at an individual level.

Estimated diffusion parameters are noisy at an individual level, especially when

the amount of data is limited. Despite this, logistic regression showed that

diffusion parameters have reasonable discriminative power. The addition of traditional

RT metrics to the classifier did not improve classification accuracy. Future work

should consider using other easily derived physiological measures (for instance

heart rate variability; E. C.-P. Chua et al. (2012)) in combination with complex

non-linear classifiers, to improve phenotypic characterization of vulnerability to

psychomotor vigilance as a result of sleep deprivation. Another way to improve

classification accuracy could be to improve the estimates of diffusion model parameters

by aggregating more data from longer periods of PVT at baseline; keeping in mind

that longer non-standard PVTs are strongly affected by time on task effects,

which might negatively affect classification. Ideally, we would like to predict such

vulnerability without having to subject participants to sleep deprivation. We will

peruse these avenues in the next chapter.

4.4.6 Strengths and limitations

One of the strengths of the present study is its large sample size. We also used

an established estimator for model parameter estimation that was validated on

simulated data. The comparison between ESD and SD while having some utility

in operational settings (Mu et al., 2005) is less ideal than a direct comparison to a

well-rested period recorded in the morning hours after waking. As a result of the

60


test protocol used in the lab, only one such morning measurement was obtained

was therefore less readily compared to the SD data (which was averaged across

two time-points (Figure 4.5).

It should be remembered that while vigilance decrements are an important and

robust measure of performance decline in sleep deprived persons, other cognitive

domains may not be similarly affected (H. Van Dongen, Baynard, et al., 2004).

Another potential limitation is that we were unable to uniquely identify all the

model parameters of the DDM.

We use the term Sleep Deprivation to refer to the interaction between homeostatic

and circadian processes instead of artificially separating the relative contributions

of the two processes. In the real world that the present work speaks to, it

remains that increased risk of vehicular accidents occurring at the nadir of the

circadian cycle after a night of sustained wakefulness correlates with slowing of

PVT performance that the diffusion model parameters in ESD predict.

0

100

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400

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600

8:00 20:00 21:00 22:00 23:00 0:00 1:00 2:00 3:00 4:00 5:00

me

an

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(m

s)

clock time

SD

RW

Figure 4.5: Mean reaction time in milliseconds across sessions. Error barsrepresent one standard error of the mean.

4.5 Conclusion

Vulnerable subjects have a lower mean diffusion drift but shorter mean non-

decision time compared to resistant group in the evening before sleep deprivation.

61


Possibly as a consequence of possessing greater cognitive reserve, resistant subjects

were less affected by SD on both decision and non-decision processes. Diffusion

drift can be used to estimate vulnerability to SD prior to experimental manipulation.

In the next chapter we will do a systematic analysis to construct a classification

system that can predict vulnerability using only baseline measures of psychomotor

vigilance. It is clear that the DDM parameters will play a crucial role in such a

system.

62

Chapter 5

Classifying Vulnerability to SleepDeprivation Using BaselineMeasures of PsychomotorVigilance

5.1 Introduction

In the previous chapter, we showed that subjects categorized as vulnerable or

resistant to sleep deprivation differed in their diffusion drift parameters (DDM)

derived from PVT RT data sampled at baseline (evening before SD to be precise).

In a recent study E. Chua et al. (2014) it was also observed that subjects who

are vulnerable to the effects of total sleep deprivation on PVT performance show

slower and more variable response times when they are well rested. These results

demonstrate that baseline PVT performance carries information about vulnerability

to subsequent sleep deprivation, but it remains unclear whether features of rested

PVT performance can be used to classify a persons relative performance in the

sleep-deprived state. As discussed earlier, one key motivation for the current work

is to harness the power of smartphones and wearable devices in the future. To

be able to predict vulnerability reliably, the classification system must work on

a widely varying experimental situation which is expected when data is collected

using these devices at the convenience of the subjects.

The analysis presented in this chapter appears in Patanaik, A., Kwoh, C. K., Chua, E. C. P.,Gooley, J. J. and Chee, M. W. L. “Classifying vulnerability to sleep deprivation using baselinemeasures of psychomotor vigilance”, Sleep, 2014. In-Press.

63

Chapter 5. Classifying Vulnerability to Sleep Deprivation Using BaselineMeasures of Psychomotor Vigilance

In this chapter we sought to assess the reliability of classifying subjects as

vulnerable or resistant to sleep deprivation, using baseline features of PVT performance.

We used two independent datasets to carry out our analysis involving PVT data

collected from different laboratories under different experimental conditions. We

extracted standard RT metrics, diffusion model parameters, and features derived

from spectral analysis of RTs, and used a support vector machine (SVM) classifier

with stratified 5 fold cross-validation (CV5) to estimate generalization error. The

objectives of this work were fourfold:

(i) Identify and rank candidate features derived from baseline PVT response

times that predict vulnerability to sleep deprivation.

(ii) Select a compact feature set from the candidate features resulting in maximum

prediction accuracy, and to measure the performance of this classification

model for each dataset.

(iii) Evaluate the reliability of classification by training the model on one dataset

and testing it on another one.

(iv) Evaluate test-retest performance of classification by training the model using

baseline data collected during one study visit, and then testing it using

baseline data collected more than 5 months later from the same set of

individuals.

5.2 Methods

5.2.1 Subjects

In this chapter we collect additional data from another laboratory. The data used

in chapter 4 (135 subjects, 69 females, age 18-25 years) constitutes Dataset 1

here. The experiments for Dataset 1 were conducted in Cognitive Neuroscience

Laboratory. For Dataset 2, 45 healthy ethnic-Chinese subjects (3 females, age

22-32 years) were enrolled in a laboratory study at the Chronobiology and Sleep

Laboratory as part of a previous study (E. Chua et al., 2014). Dataset 2 shared

the same recruitment criteria as Dataset 1 which is discussed in detail earlier

(Section 4.2.1). Informed consent was obtained from all participants, and research

procedures were approved by the SingHealth Centralized Institutional Review

Board.

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5.2.2 Sleep deprivation procedures

Dataset 1: As discussed earlier, subjects arrived in the laboratory in three sessions.

In this chapter we are specifically interested in the sleep deprivation session, in

which the subjects arrived at the laboratory at 7:30pm and were kept awake

continuously overnight under supervision of a research assistant. A hand-held 10-

min PVT was administered every hour from 8:00pm to 5:00am (10 test periods).

Subjects were seated upright during testing and were exposed to ordinary room

light. Participants movements were not restricted between PVT tests.

Dataset 2: Subjects underwent total sleep deprivation in a laboratory suite

that was shielded from external time cues. Participants arrived in the evening and

went to bed at their regular pre-study sleep time. After 8-hours of time in bed

for sleep, subjects were kept awake for at least 26 hours using constant routine

(CR) procedures, as previously described (Duffy & Dijk, 2002). During the CR

procedure, subjects remained in bed in a semi-recumbent position, with exposure

to dim ambient lighting (< 5 lux). The PVT was administered every two hours

(starting 2.5 h or 4.5 h after wake time) by computer using E-Prime 2 Professional

software (Psychology Software Tools, Inc., Sharpsburg, PA). Visual stimuli were

presented on an LCD monitor placed on an over-bed table, which allowed subjects

to take the PVT while remaining in bed. After undergoing sleep deprivation,

participants were invited to return to the laboratory at least 5 months later to

complete additional testing. A subset of subjects (n=34) took part in the follow-

up study. Subjects reported to the laboratory in the mid-afternoon (between 2

6 pm) and completed two 10-min PVTs taken 2 h apart from one another under

conditions that were similar to the first study visit. The research protocols for

both datasets are summarized in Figure 5.1.

5.2.3 Assessment of vulnerability to sleep deprivation

During total sleep deprivation, cognitive performance usually reaches its nadir

in the early morning hours, typically between 4am and 8am, when the sleep

homeostat and circadian clock interact to promote high levels of sleepiness (Achermann

& Borbely, 1994). This is also the period when sleepiness-related motor vehicle

accidents are most likely to occur (Horne & Reyner, 1999). We therefore analyzed

PVT performance during this time window as a measure of susceptibility to sleep

deprivation. Subjects were categorized as vulnerable or resistant based on a

65


Original

24 4 8 12 16 20 24 4 8

Sleep Sleep

baseline

Clock time (h)

SD

A

B

Follow-up

baseline

24 4 8 12 16 20 24 4 8

Clock time (h)

Sleep deprivation Session

Sleep Sleep

baseline SD

> 5 months

n=45

n=34

n=135

Figure 5.1: (A) Dataset 1: Subjects (n=135) arrived at the laboratory at 7:30pmand stayed awake overnight resulting in total sleep deprivation of 22 h. A 10-min PVT was administered every hour from 8:00pm until 5:00am on the nextmorning. (B) Dataset 2: After an 8-h opportunity for sleep, subjects underwentsleep deprivation in the laboratory for at least 26 h. Every 2 h, subjects completeda 10-min psychomotor vigilance task (PVT), indicated by the circles. A subset ofsubjects (n=34) participated in a follow up session in which two PVTs were takenin the mid-afternoon. In each dataset, subjects were stratified into vulnerableand resistant groups by performing a median split of PVT lapse data (reactiontimes > 500 ms) during the last session of sleep deprivation (red circles). Twobaseline PVT sessions (green circles) were used to build the classifier for predictingvulnerability to sleep deprivation.

median split on the number of lapses, defined as RTs that exceed 500 milliseconds

(Figure 5.1. PVTs marked in red). For Dataset 1, 70 subjects were categorized as

vulnerable (≥5 lapses). For Dataset 2, 25 subjects were categorized as vulnerable

(≥23 lapses), of whom 19 completed the follow-up study. The large difference

in the median lapse value is likely a result of different experimental conditions

between the two studies. The divergence in performance between vulnerable and

resistant groups after their usual bedtime is shown in Figure 5.2. For Dataset

1, the first two PVT tests administered at 8:00pm and 9:00pm were used as

the baseline. For Dataset 2, the third and fourth PVT sessions, which were taken

during the mid-afternoon, were used as the rested baseline, as PVT measurements

for the follow up session were available for the same time period.

66


0

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pse

s

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Figure 5.2: Time course of PVT lapses in vulnerable and resistant groups for (A)Dataset 1 and (B) Dataset 2. Inset: Individual traces show the time course oflapses for each participant who underwent sleep deprivation. Mean ± SEM areshown.

5.2.4 RT Derived Features

As detailed below, we used a combination of standard RT metrics, features derived

from the drift diffusion model (DDM), and spectral analysis of RTs.

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5.2.4.1 Standard RT metrics

The most widely used PVT outcome metric is the number of lapses followed by

mean RT, mean 1/RT, fastest 10% RT, median RT, slowest 10% RT, and the

slowest 10% 1/RT (Basner & Dinges, 2011). The reciprocal RT is also referred

to as response speed (RS=1/RT). In an analysis of various PVT outcome metrics

during partial or total sleep deprivation, Basner & Dinges (2011) found that

metrics involving response speed (RS) and lapses were the most sensitive to sleep

loss. It was therefore recommended that these measures be used as the primary

outcome measures of the 10-min version of the PVT. In our analysis we considered

lapses, mean RT, mean RS, slowest 10% RS, fastest 10% RT, median RT and

standard deviation of RT. We also included mean absolute deviation (MAD) from

the mean, and ∆RT > 250, which is the number of consecutive RTs that differ

by more than 250ms. The latter was included based on a previous finding that,

under baseline conditions, subjects who were categorized as vulnerable to sleep

deprivation showed a greater number of consecutive RTs that differed more than

250ms compared to resistant individuals (E. Chua et al., 2014).

5.2.4.2 Metrics derived from DDM

We included the standard normalized drift parameters: mean diffusion drift ξ/a,

variability in mean diffusion drift η/a, mean non-decision time Ter and range of

non-decision time st. Again, we implicitly assume that when we refer to the drift

parameters they are normalized by the boundary parameter. We also included

the parameter ratio ξ/η which is the diffusion drift signal to noise ratio (SNR).

This parameter is known to closely track alertness (Ratcliff & Van Dongen, 2011).

While estimating the parameters of the model, we combined RT data from two

consecutive PVT sessions taken during the baseline rested state.

5.2.4.3 Metrics derived from Spectral analysis of RTs

Given that spatio-temporal features (i.e. structures in the ordering and positioning

of the RTs) might not be captured by standard PVT metrics or the DDM, we used

discrete wavelet transform (DWT) to extract multi-resolution features from RTs.

The DWT effectively addresses the tradeoff between time and frequency resolution

in signal analysis and can handle non-stationary signals as well. It decomposes

the signal into a hierarchical set of low frequency and high frequency components

68


called approximations and details respectively. The DWT is computed by successive

low pass and high pass filtering in the time domain. First the samples of RTs,

r[n] are passed through a low pass filter with impulse response g[n] resulting in

a convolution of the two signals. The signal also goes simultaneously through

a high pass filter with impulse response h[n]. The outputs then give the detail

coefficients (d[n] from high pass filtering) and the approximate coefficients (a[n]

from low pass filtering). The filter outputs are then subsampled:

yhigh[n] =∞∑

k=−∞

r[k] ∗ h[2n− k] (5.1)

ylow[n] =∞∑

k=−∞

r[k] ∗ g[2n− k] (5.2)

The process is applied repeatedly to get multi-level decomposition at different

scales (Figure 5.3). We used the Haar wavelet to construct the filter bank. Due to

the way DWT operates, the length of the signal has to be a power of 2. The number

of RTs collected per PVT can vary widely depending on experimental conditions.

The only way to handle the signal is to either truncate them or to extend them.

To avoid introducing undesirable artifacts due to signal extensions and maintain

consistency across subjects and datasets we considered the last 26 = 64 samples

of RTs from each PVT. In the baseline condition, lapses were rare enough and

subjects were fast enough to guarantee 64 samples. Samples from two PVTs in

the rested baseline were combined together (resulting in uniform sample size of

128) and a 6 level DWT was applied. For each level l the Mean Absolute Value

(MAV) was computed as:

MAVl =1

N

N∑n=1

|dnl | (5.3)

where dnl is the nth detail coefficient at level l. RT data was demeaned before

application of wavelet transform.

5.2.5 Feature Selection

In any supervised classification problem, identifying the essential features is critical

to the performance of the classifier. Some of the features might be irrelevant in the

sense that they provide no additional information from the point of view of class

69


Figure 5.3: A four level decomposition of signal using discrete wavelet transform.At each level, the resolution of the signal is halved.

prediction. Features could also be redundant, i.e. in presence of other relevant

features they provide no additional information. Moreover the irrelevant and

redundant features increase computational complexity and might introduce noise

into the system and reduce the performance of the classifier. Therefore, given a

set of features F = fi, i = 1, ...,M existing in RM and the target class variable

c, feature selection tries to find a subset S ⊂ F with m features that optimally

characterizes c. Ideally we would like to minimize the generalization error of the

classifier. The global solution, which might not be unique, could be found by

exhaustively searching the feature space. This would require 2M − 1 operations.

Unfortunately this becomes computationally infeasible even for moderately large

values of M . Additionally, since the generalization error has to be estimated from

data, large number of searches increases the chances of over fitting especially when

the data size is small.

For our analysis, we first eliminated highly redundant or irrelevant features to

focus on a small critical set of candidate features based on minimal-redundancy-

maximal-relevance (mRMR) (Peng et al., 2005) for both datasets independently.

The mRMR criterion subtracts the minimal redundancy from maximum relevance

and can be expressed as:

1

|S|∑fi,∈S

I(fi, c)−1

|S|2∑

fs,fi∈S

I(fs; fi) (5.4)

Where, I(X, Y ) is the mutual information (MI) between random variable X and

Y and can be computed as:

I (X, Y ) =

∫ ∫p (x, y) log

(p (x, y)

p (x) p (y)

)dxdy (5.5)

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We estimated the MI using kernel density based estimation (Bowman & Azzalini,

1997). We used normal kernel function evaluated at 100 equally spaced points.

The candidate feature set was constructed by using a first-order incremental search

that maximizes the mRMR criterion at each stage. It was observed that the

classification accuracy stabilized by the 8th ranked feature for both datasets, when

each feature was considered incrementally. Therefore, only the top 8 features were

selected as the candidate feature set for further analysis.

Even though a selected candidate feature is deemed relevant, including all

such features does not necessarily result in better classification rate as compared

to using a smaller feature set. Hence, relevance does not imply optimality (Kohavi

& John, 1997). As a result the m best features are not the same as best m features

(Cover, 1974). To maximize the accuracy of the classifier, a further compact

subset of features was selected from the candidate features using a wrapper based

approach (Kohavi & John, 1997). A wrapper is a feature selector that is wrapped

with the classifier to select features that result in the lowest generalization error.

This is very similar to the naive method described earlier, but used on a much

smaller candidate feature set employing some heuristic schemes instead of trying

out all possible combinations of features. We employed both the incremental

forward and backward selection wrapper schemes detailed previously (Kohavi &

John, 1997).

5.2.6 Classifier

For classification we have considered support vector machines (SVMs) with radial

basis function (RBF) kernel (Vapnik, 2000). SVMs are powerful supervised classification

methods with strong theoretical foundations in statistical learning theory and

structural risk minimization; thereby leading to good generalization. A standard

SVM tries to find a separating hyperplane between two classes that maximizes

the margin. For data that is not linearly separable a kernel function is used

that transforms the original data to a high dimensional (infinite dimensional

in the case of RBF kernel) space, where the data is linearly separable. For a

detailed discussion please refer to (Vapnik, 2000). To estimate the generalization

error, a stratified 5-fold cross validation (CV5) was used. To find the best SVM

parameters a broad level grid search was employed (regularization parameter

71


C = [20, 22, . . . , 210] and RBF kernel parameter γ = [20, 21, . . . , 25]). As a pre-

processing step, features were normalized to unit hypersphere before feeding them

to the classifier.

f =f

||f ||(5.6)

Where f is the normalized feature vector and ||.|| is the `2 norm. All analysis

were implemented in Matlab 2013b, The MathWorks, Inc., Natick, Massachusetts,

United States. We used Matlab version of LIBSVM (Chang & Lin, 2011) in our

analysis. The complete feature selection process is summarized in Figure 5.4.

Rested ConditionSleep Deprived

Condition

PVT PVT

Sleep Deprivation

RTs RTs

Median split on lapses

vulnerable

resistant

Response Category

DDM Wavelet Summary Statistics

Max Relevance Min Redundancy

Ranked featuresTop 8 features

Wrapper heuristics using CV5 SVM

Best set of features

Figure 5.4: Summary of the overall feature selection process. The process wasapplied independently on both datasets.

5.2.7 Statistical Analyses

Standard RT metrics are known to be stable and reproducible across studies

(E. Chua et al., 2014). To test the reproducibility of the DDM parameters,

parameters estimated from baseline data of Dataset 1 collected in the first visit

72


were compared with parameters estimated from the follow up study using two way

repeated measures ANOVA with group (vulnerable vs resistant) as the between

subject factor and schedule (first visit versus follow up) as the within subject

factor.

The variation in DDM parameters for vulnerable and resistant subjects on the

evening before and the morning after sleep deprivation has been reported in the

previous chapter. As PVT measurements for Dataset 2 were available across the

day, DDM parameters were also estimated for the vulnerable and resistant groups

at different time of the day. This was also done using repeated measures ANOVA

(group as between subject and time of day as within subject factor, with data

binned for two sessions of the PVT). For ANOVAs with statistically significant

interaction, post-hoc t-tests were used to examine simple effects of group and

time. The classifiers were compared using McNemar test with Yates correction.

Statistical analyses were performed using SPSS (IBM Corp., New York, NY).

Statistical significance was set at α = 0.05.

5.3 Results

In each dataset, the top 8 features ranked according to their mRMR score are

shown in Figure 5.5. As described in the methods, a wrapper based approach was

used to select a subset of features that minimized the generalization error. Below

we report classifier accuracy as assessed by cross validation (CV5 accuracy) of

features using the optimal feature set.

5.3.1 Model performance in Dataset 1

A CV5 accuracy of 77% was obtained with sensitivity of 85.7% and specificity

of 67.7% using five features: diffusion drift ξ, variability in diffusion drift η,

range of non-decision time st, the number of consecutive RTs that differed by

more than 250ms (∆RT > 250) and wavelet feature MAV6 (Figure 5.6A). The

receiver operating characteristic (ROC) curve and associated confusion matrix are

presented in (Figure 5.6A). The area under the ROC curve (AUC) is often used

as a single-value representation of overall classifier performance and is equivalent

to the probability that the classifier will rank a randomly chosen positive instance

higher than a randomly chosen negative instance. For Dataset 1, the AUC for the

classifier was 0.75.

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5.3.2 Model performance following training on Dataset 1and testing on Dataset 2

We trained the model on Dataset 1, using the corresponding five features deemed

most optimal and applied this feature set to Dataset 2. An accuracy of 71.1% was

obtained with sensitivity of 80% and specificity 60%. On varying the threshold

for the classifier, a more balanced result was obtained with sensitivity of 72%

and specificity of 70% while the accuracy remained unchanged. The ROC and

confusion matrix at the default threshold as well as the balanced threshold are

presented in Figure 5.7. AUC for the classifier was 0.73.

5.3.3 Model performance in Dataset 2

For Dataset 2, diffusion drift ξ, range of non-decision time st and the number of

consecutive RTs that differed by more than 250ms (∆RT > 250) were deemed

most important. A CV5 accuracy of 82.2% was obtained with sensitivity of 84%

and specificity of 80% using three features: diffusion drift ξ, range of non-decision

time st and the number of consecutive RTs that differed by more than 250ms

(∆RT > 250). The receiver operating characteristic (ROC) curve and associated

confusion matrix are presented in Figure 5.6B. For Dataset 2, the AUC for the

classifier was 0.74.

5.3.4 Reproducibility of classification across testing episodesin the same participants

Using the optimal feature set selected in Dataset 2, when the model was trained

on baseline PVT data collected during the first visit and then tested on PVT

data from the follow-up visit, we obtained an accuracy of 79.4%, with sensitivity

of 73.7% and specificity of 86.7%.

5.3.5 Classification using the most sensitive PVT measures

Although standard PVT metrics were not selected by the model, we considered

the possibility that such measures might nonetheless carry information about

vulnerability to sleep deprivation. Using lapses, mean RS and slowest 10% RS

measured at baseline, three measures previously found to be most sensitive to

total sleep deprivation32, we obtained a CV5 accuracy of 64.4% on Dataset 1

74


and CV5 accuracy of 68.9% on Dataset 2. Compared to the best performing

features for each dataset, the accuracy of classification using these PVT measures

was significantly poorer in the case of Dataset 1 (χ21 = 10.2, P < 0.005) but not

significantly different for the smaller Dataset 2, (χ21 = 1.8, ns).

5.3.6 DDM variability across visits and with time; computationalperformance

Except for across-trial variability in drift η (F1,32 = 4.65, P < 0.05), DDM

parameters were stable across study visits (Figure 5.8). Vulnerable subjects

had lower mean drift (F1,32 = 11.6, P < 0.005) and higher variability in non-

decision time (F1,32 = 13.6, P < 0.001) across both visits compared to resistant

subjects. Although DDM parameters were significantly affected by time of the day

(Figure 5.9), vulnerable subjects had lower mean diffusion drift (F1,43 = 4.03, P <

0.001) and higher variability in non-decision time (F1,43 = 27.5, P < 0.001)

irrespective of time elapsed since wake.

The feature extraction process was reasonably quick (∼ 10 min /subject)

when run on a modern workstation, with the majority of time spent estimating

the DDM parameters. Overall, the χ2 DDM fit (at baseline) for Dataset 1 was

19.2 ± 6.1 and for Dataset 2 it was 16.8 ± 7.2. The critical value for χ2, df=14

was 26.1.

75


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ccu

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B

Figure 5.5: Stratified 5-fold cross validation (CV5) accuracy of the classifieras features ranked by minimal-redundancy-maximal-relevance (mRMR) criterionwere incrementally added to the feature set for (A) Dataset 1 and (B) Dataset 2.The top 8 features in each dataset (partitioned by vertical line) were consideredfor the candidate feature set. The feature set comprised of 5 DDM parameters(mean drift: ξ/a, across trial variability in drift: η, mean non-decision time:Ter, variability in non-decision time: St, drift signal to noise ratio: driftSNR); 9standard reaction time (RT) metrics (mean RT, mean response speed (RS), fastest10% RT, median RT, slowest 10% RS, median RT, standard deviation of RT: stdRT, mean absolute deviation of RT: MAD RT and the number of consecutive RTsthat differ by more than 250ms: ∆RT > 250); and 6 metrics derived from spectralanalysis of RTs (Mean Absolute Value of detail coefficient at level 1 through level6: MAV1 to MAV6).

76



TP

Ra

te (

Se

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Accuracy: 77%

Specificity: 67.7%

Sensitivity: 85.7%

AUC: 0.75

predicted

actual

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Se

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tivit

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Specificity: 80%

Sensitivity: 84%

AUC: 0.74

predicted

actual

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Figure 5.6: Receiver operating characteristic (ROC) curves obtained by varyingthe threshold of class membership probability of the SVM classifier for (A) Dataset1 and (B) Dataset 2 using the best set of features. The best performing point onthe ROC curve is marked with a gray circle. Inset: confusion matrix, accuracy,sensitivity and specificity at the best performing point.

77



TP

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predicted

actual

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00

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resistant 12 8 20

vulnerable 5 20 25

17 28 45

Accuracy 71.1%

Sensitivity 80%

Specificity 60%

Default threshold


resistant 14 6 20

vulnerable 7 18 25

21 24 45

predicted

actualAccuracy 71.1%

Sensitivity 72%

Specificity 70%

Figure 5.7: Performance of the classifier trained on Dataset 1 and tested onDataset 2 is demarcated by the gray circle on the receiver operating characteristic(ROC) curve. The corresponding confusion matrix, accuracy, sensitivity andspecificity are presented below. A more balanced classifier performance wasobtained by changing the class membership probability threshold from its defaultvalue (marked with a black circle on the ROC). While the accuracy did notimprove, the sensitivity and specificity became more balanced.

78


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Figure 5.8: Estimated drift diffusion model (DDM) parameters for vulnerableand resistant subjects estimated from baseline PVT sessions measured acrosstwo study visits for Dataset 1. Of the 45 subjects who participated in the firststudy, 34 returned for a follow up study after at least 5 months following theirinitial visit to the laboratory. Baseline individual differences in mean diffusiondrift and variability in non-decision time were reproducible across study visits.Vulnerable subjects had lower mean drift (F1,32 = 11.6, P < 0.005) and highervariability in non-decision time (F1,32 = 13.6, P < 0.001) across both visitscompared to resistant subjects. The across trial variability in diffusion driftappeared to be the only DDM parameter to be affected across the two studies(F1,32 = 4.65, P < 0.05). The ‘#’ symbol denotes significant main effects ofgroup on DDM parameters. Mean ± SEM are shown.

79


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Figure 5.9: Estimated drift diffusion model (DDM) parameters for vulnerableand resistant subjects across the study period for Dataset 2. DDM parameterswere significantly affected by time of day. Vulnerable subjects had lower meandiffusion drift (F1,43 = 4.03, P < 0.001) and higher variability in non-decisiontime (F1,43 = 27.5, P < 0.001) irrespective of time elapsed since waking. Acrosstrial variability in drift and mean non-decision time showed interesting variationsover the period of the day. The combined effect of all the DDM parameters wassuch that depending on the time of the day, the parameters could act in oppositedirections. The dotted vertical line demarcates the usual bed time. Asteriskindicates statistically significant difference (P < 0.05).

80


5.4 Discussion

We observed large between-subject differences in PVT performance during sleep

deprivation, such that sleep deprivation vulnerable persons had on average 3-8

times more lapses than members of the resistant group. Although prior studies

have identified baseline differences between individuals who are either vulnerable

or resistant to sleep deprivation, the ability to predict performance vulnerability

using baseline data has not been systematically examined. Here, we examined

features beyond summary statistics conventionally used in assessing RTs, including

measures derived from the DDM and spectral analysis. Using two independent

datasets, we identified a subset of PVT features that can be used to classify

relative vulnerability to total sleep deprivation with about 77-82% accuracy.

5.4.1 Features most useful for discriminating vulnerableand resistant participants

Despite substantial differences in the way that PVT data were collected across the

2 studies considered here, DDM parameters and wavelet MAV parameters were

among the top baseline features associated with relative vulnerability to sleep

deprivation. In terms of best performing features, three features were selected

in both datasets: diffusion drift ξ, range of non-decision time St and the no. of

consecutive RTs that differ by more than 250 ms (∆RT > 250). Interestingly,

none of the standard PVT performance metrics (e.g., mean RT and lapses) were

selected when the mRMR criterion was used. The empirical data collected suggest

that relative to standard RT metrics, the DDM better captures useful information

embedded in RT data that can distinguish persons vulnerable to vigilance decline

following sleep deprivation by decoupling RT into distinct components. This

is consistent with our previous findings that diffusion parameters measured at

baseline predicted vulnerability despite the absence of significant differences in

baseline standard RT metrics. This might not be surprising upon inspecting the

variation of the DDM parameters for the two groups across the day for Dataset

2 (Figure 5.9). While the mean diffusion drift and range of non-decision time

showed statistically significant differences across groups irrespective of time of

day, the mean non-decision time and across trial variability parameters showed

interesting variations throughout the day. Depending on the time of the day,

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DDM parameters moved in opposite directions; i.e. some DDM parameters had a

tendency of increasing the RT while others had a tendency to decrease it. In other

words, the decision and non-decision components can trade off with each other

without affecting overall observed performance. This has also been demonstrated

in the previous chapter using simulation (see Figure 4.4).

5.4.2 Classification reliability and reproducibility

PVT outcomes can be affected by multiple factors including experimental conditions,

interventions and time of day. Here, we showed that the same set of features

appear to be highly discriminatory across datasets. For Dataset 1, five features

were selected by the model: diffusion drift ξ, variability in diffusion drift η, range

of non-decision time st, ∆RT > 250 and MAV6. A CV5 accuracy of 77% was

achieved using these features. Despite differences in experimental conditions and

baseline data acquisition times between the two datasets, the model trained on

Dataset 1 and tested on Dataset 2 showed only a small drop in classification

performance (71% accuracy). Importantly, the model performed well despite the

large difference in the average number of lapses between studies, i.e., the median

split on Dataset 1 cannot be directly linked to the median split on Dataset 2.

Allowing the threshold to change when the model trained on Dataset 1 was applied

to Dataset 2 resulted in a more balanced classification. These results attest to

the utility of our classification model for predicting relative vulnerability to sleep

deprivation.

Due to aforementioned dataset differences, it might be expected that the best

set of features selected by the model would be dissimilar across the two datasets.

However, we found that of the five features assessed as most optimal for Dataset

1, three were again selected for Dataset 2 (diffusion drift ξ, range of non-decision

time st and ∆RT > 250). Using these features, CV5 accuracy of 82.2% was

achieved. Importantly, when the classifier was trained on PVT data collected

during baseline of the first study visit in Dataset 2, performance of the model was

similar to that for data analyzed during the follow-up visit (79.4% accuracy). Since

the second visit occurred more than 5 months after sleep deprivation, our results

suggest that baseline diffusion drift ξ and range of non-decision times st show

stable between-subjects differences over time. While the best set of features might

change depending on the type of classifier employed, the type of heuristic used, and

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the definition used for categorizing subject vulnerability, our findings nonetheless

demonstrate that DDM parameters, wavelet parameters, and ∆RT > 250 are

useful for predicting vulnerability to decline in psychomotor vigilance following

total sleep deprivation under laboratory conditions.

5.4.3 Differences between datasets

In Dataset 1, participants were free to walk around when not taking the PVT, and

they were exposed to ordinary room light. In Dataset 2, PVT data were collected

under conditions that were conducive to poorer performance. Specifically, subjects

were restricted to bed in a semi-reclined position in constant dim light. Also, in

Dataset 2 we used baseline PVT data collected in the late afternoon, corresponding

to the mid-afternoon dip in performance. By comparison, baseline PVTs in

Dataset 1 were taken in the late evening, corresponding to the wake maintenance

zone when the circadian drive to remain awake is near its peak. The aforementioned

experimental differences may explain, in part, the slightly better classification

rates in Dataset 2 using a smaller number of features.

5.4.4 Definition of vulnerability to total sleep deprivation

In our analysis, group assignment was based on a median split using the number of

lapses in the last session of sleep deprivation. The number of lapses (RTs>500ms)

is the most commonly used PVT metric to assess the effects of sleep deprivation on

sustained attention. We acknowledge, however, that vulnerability could be defined

using other PVT outcome measures, e.g., response speed, or by individualizing

the lapse threshold relative to each persons baseline performance. Dividing each

dataset into two groups enabled us to build a model that predicts a binary

outcome (resilient or vulnerable), but it is important to note that vulnerability

to sleep deprivation is a continuous variable. Hence, the decision to split the

dataset by the median was arbitrary, and the most vulnerable resistant subjects

were qualitatively similar to the most resistant vulnerable subjects, separated

only by a few PVT lapses during sleep deprivation. Despite this limitation,

subjects were classified with nearly 80% accuracy. Of note, although resistant

and vulnerable groups differed by a few lapses even during the baseline rested

state (Figure 5.2), the feature selection process did not select lapses in either

dataset. Instead, vulnerability was better predicted by underlying decision and

non-decision components of the DDM.

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5.4.5 Further improvements in classification

The reliability of estimates for DDM parameters is affected by the size of the

data set as has been demonstrated in Chapter 3. To ensure that we had sufficient

data for the DDM, we combined PVT RT data across two consecutive 10-min

PVT sessions, separated by an hour (Dataset 1) or two hours (Dataset 2). DDM

parameter estimates might be further improved by implementing longer PVT

sessions (e.g. 20-30 min in duration) or by aggregating data across more PVT

sessions. It must be kept in mind, however, that longer-duration PVTs are more

affected by time-on-task effects (H. Van Dongen et al., 2011), and adding more

PVT sessions might not be practical if our model is to be applied in real-world

settings. The classification might also be more accurate if participants are studied

under baseline conditions that are conducive to sleep. Adding other easily derived

physiological measures, for instance heart rate variability (E. C.-P. Chua et al.,

2012), could also potentially improve the performance of the classifier.

5.5 Conclusion

In this chapter, we built a classifier to predict vulnerability in sustained attention

during sleep deprivation, using features derived from PVTs taken under rested

baseline conditions. We included a range of features including several summary

statistics, DDM parameters, and features derived from wavelet transform of the

RT sequence. We found that DDM parameters, including decision and non-

decision components, the number of consecutive RTs that differ by more than

250ms and wavelet features can be used to discriminate sleep deprivation vulnerable

and resistant individuals with an accuracy of 77-82% across datasets. Now that we

have shown that there are baseline differences between vulnerable and resistant

subjects, and these differences are enough to classify the subjects even under

widely varying experimental conditions, in the next chapter we will explore if

these differences reveal themselves in resting state conductivities prior to sleep

deprivation.

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Chapter 6

Baseline resting stateconnectivity differences betweenindividuals vulnerable andresistant to sleep deprivation

6.1 Introduction

In this chapter we turn our attention to neuroimaging studies. Specifically we

are interested in knowing if differences in functional connectivity in the brain

are observable in the resting brain prior to SD. We have shown in the previous

chapters that such differences are observable in the DDM parameters prior to SD,

suggesting intrinsic differences in the structural and/or functional brain. Several

studies have also demonstrated baseline neurophysiological differences between

participants vulnerable and resistant to sleep loss that could either be detected

before exposure to SD or represent a stable trait of the brain. For instance, Mu et

al. (2005) showed that participants more vulnerable to SD on a Sternberg working

memory task displayed reduced global brain activations than when they were

well-rested. In addition, Rocklage et al. (2009) observed differences in distributed

white matter (WM) pathways between vulnerable and resistant participants on

a simple visual motor task. Although, only a few studies exists (De Havas et

al., 2012; Gujar et al., 2010; Samann et al., 2010) that have examined functional

The analysis presented in this chapter appears partly in Patanaik, A. and Zagorodnov, V.“Connectivity between visual resting state networks predicts vulnerability to sleep deprivation.”,Organization of Human Brain Mapping, 2012. Used with permission.

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Chapter 6. Baseline resting state connectivity differences between individualsvulnerable and resistant to sleep deprivation

connectivity differences between rested condition and sleep deprivation. There are

no studies that examine these differences between vulnerable and resistant groups.

Before we continue, would provide some background to the current investigation.

6.2 Preliminaries

6.2.1 BOLD - fMRI

When neurons in the brain become active, the blood flowing through that area

increases. Furthermore the amount of blood that is sent to that area is more than

that is required to replenish the oxygen that is used up by the cells. This leads

to a local increase in blood oxygenation level. Functional Magnetic Resonance

Imaging (fMRI) measures this change in oxygenation level which is therefore

referred to as blood oxygenation level dependant (BOLD) fMRI. It is important to

notice that fMRI does not measure neuronal activity directly, rather it measures

the BOLD signal which acts like a proxy for neuronal activity. This increase in

blood flow following brief period of neuronal activity is known as hemodynamic

response. Hemodynamic response is slow; whereas neuronal activity typically

lasts for milliseconds the increase in blood flow following it may take as much

as 5 seconds to reach its maximum. Furthermore the fall (or undershoot) of

the hemodynamic response is very slow taking as much as 15 to 20 seconds to

reach baseline levels. Fortunately the hemodynamic response can be modelled

satisfactorily by a linear time invariant system. This allows for construction of

hemodynamic responses from expected neuronal response using convolution. This

is very useful in experimental design to find locations in the brain that activate

(or deactivate) with a particular experimental task.

6.2.2 Data Acquisition

The data acquisition occurs slice by slice, usually in an alternate order i.e. from top

to bottom for one scan and bottom to top for another. The time interval between

two scans is called repetition time (TR) which decides the temporal resolution of

the data. TR along with in place resolution and slice thickness decides the overall

spatial and temporal resolution (Figure 6.1). It is possible to trade off temporal

resolution for spatial resolution. The researcher has to decide the parameters

based on the aims of the experiment.

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Figure 6.1: fMRI data acquisition process. The spatial resolution of the scan ish× w × d and temporal resolution is TR.

6.2.3 Data Pre-Processing

Before the acquired data can be analyzed, it must go through some standard pre-

processing steps. Slice time correction and motion correction are usually the ones

performed first. Slice time correction (Henson et al., 1999) is required because

each slice is taken at a different time. Different interpolation algorithms like linear

or sinc (sin(x)/x) interpolation can be used to correct for different slice acquisition

time, though these are based on the assumption that subject is not moving, which

is not true. Unfortunately, any motion correction algorithm must assume that

images are slice time corrected! Due to such interdependencies of assumption

between slice time correction and motion correction there is no consensus as to

which one should be done first. Experience suggests that for short TRs (< 2

sec) slice time correction is not required. Also new MRI machines can acquire

interleaved slices, which reduce the need for slice-time correction further.

Next in the pre-processing pipeline is spatial normalization which includes

coregistration and normalization. While at the gross level human brain shows

remarkable consistency in its overall structure across individuals, it varies widely in

size and shape. This makes spatial normalization a necessary step. Coregistration

is the process of aligning functional images to high resolution structural image

to help identify specific anatomical structures in the functional images. Next

the functional image is normalized to a standard space. The Talairach space

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(Talairach & Tournoux, 1988) and the Montreal Neurological Institute (MNI)(Evans

et al., 1992) space are the two most widely used spaces in the neuroscience

community. The normalization process can be achieved using landmark based

methods, volume based methods and surface based methods using linear, affine

or non-linear registration algorithms. The complete discussion of these is out of

scope of this thesis, interested readers are referred to (Holden, 2008). In the last

step spatial and temporal smoothing are applied. Spatial smoothing is necessary

to increase signal to noise ratio (SNR) to acceptable levels. It also allows for

slight mismatch across individuals at the expense of spatial resolution. Temporal

smoothing may be required to remove un-interesting temporal signals from the

data. Temporal filtering is normally done in frequency domain. For a more

detailed discussion of the fMRI processing pipeline reader is referred to (Poldrack

et al., 2011) which gives a clear and concise description of fMRI data analysis.

6.2.4 General Linear Model-Finding regions of significanttask activations

In task based fMRI analysis the objective is to find brain regions that are significantly

activated with the task. This is achieved by entering the task and individual

voxel BOLD response into a general linear model (Figure 6.2). The task is

normally convolved with hemodynamic response function (HRF) which is the

impulse response of the hemodynamic system. In the figure, β1 accounts for the

variability in BOLD signal explained by the task. β2 accounts for the baseline.

The unexplained variance is accounted for by the error. The GLM runs on each

of the voxels at the end of which parametric maps are obtained corresponding

to each voxel. Statistical significance tests are then performed on these maps to

get statistical parametric maps (SPMs). SPM is an image wherein the intensity

value represents statistical significance obtained under the null hypothesis of no

activation. These SPMs are then thresholded to obtain regions of significant task

activations.

6.2.5 Statistical ThresholdingProblem of Multiple Comparisons

The easiest way to threshold is to declare a voxel activated if the corresponding

SPM intensity exceeded a predefined threshold α derived from null distribution.

This is similar to any statistical significance test with significance level α with

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BOLD Signal Task waveform * HRF Baseline error

= β1 + β2 +

Figure 6.2: To find regions of significant task activation GLM is used to find outthe variability in measured signal explained by the task. Statistical tests are doneon the coefficients β to find voxels with significant task activation.

one key difference; In case of SPM this significance test is done for each voxel.

Given that a standard SPM will have in the order of ≈ 105 voxels, huge number

of voxels will turn up activated just by chance even for most conservative levels

of significance. For e.g. a standard SPM of 64 ∗ 64 ∗ 32 = 131, 072 voxels with

a conservative α = 0.001 may show up an expected 131 voxels as activated just

by chance! In this situation, false positives (type I error) must be controlled over

all tests, but there is no single measure of Type II error in such multiple testing

problems (Hochberg & Tamhane, 1987). The chance of any Type I error over

whole dataset is the family wise error (FWE) rate. Bonferroni correction can be

applied to control the FWE, this would come at the expense of increasing type II

error (missed detections or false negatives). Unfortunately, for most practical

fMRI applications Bonferroni correction is very conservative and the reduced

power of the test makes it useless. Even methods which assume independence

of individual tests (like Dunn-idk correction) do not work well because fMRI

data has implicit smoothness in it which makes the assumption of independent

tests invalid. Another way of handling the problem of multiple comparisons in

the context of fMRI data is to incorporate spatial information by testing voxel

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clusters rather than individual voxels based on theory of Random Fields (Poline et

al., 1997). While useful, Random Fields Theory (RTF) methods have limitations

mainly caused by violations of the assumptions it makes (see (T. Nichols &

Hayasaka, 2003)), also it requires estimation of smoothness in the data which

may not always be reliable. This is where non-parametric permutation tests

(T. E. Nichols & Holmes, 2002) come into picture. Permutation tests do not

make any assumptions about the data distributions; instead they allows one to

directly obtain the null distribution of non-activated voxels. They works under

the assumption of exchangeability under the null hypothesis. This is intuitively

easy to understand. Take for example, two groups A and B under the null

hypothesis that the mean of the two groups are same. If the null is true then data

exchangeability holds and data from group A can be exchanged by data in group

B. By randomly permuting data between the two groups the distribution of the

null can be obtained, statistical tests can then be easily performed. The problem

of multiple comparisons in the context of permutation tests is easy to solve using

maximal summary statistics over all the voxels. The reason permutation tests were

not popular until recently is their heavy computational cost. But with increased

computational power at hand, permutation tests are very powerful and should be

preferable to other methods.

Cluster based analysis is generally preferred as it can be more sensitive to

finding true signal than voxel wise thresholding (see for e.g. K. J. Friston et al.

(1996)). No matter which method is used, to carry out cluster based analysis

instead of voxel based one must first define the clusters. The cluster threshold is

normally arbitrary, yet its exact choice can have significant impact on the results.

Threshold free cluster enhancements (TFCE) (S. M. Smith & Nichols, 2009) is

a relatively new method which addresses this issue. It implicitly relies on non-

parametric permutation tests and is shown to be better than conventional methods

especially under low SNR conditions.

6.2.6 Functional Connectivity and Brain Networks

Brain regions do not work in isolation, but rather organize themselves into set

of connected networks. This connectivity could be anatomical, functional or

effective. Anatomical connectivity is direct axonal connection between brain

regions which can be established by diffusion tensor imaging (DTI) studies. Functional

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connectivity (K. Friston, Frith, Liddle, & Frackowiak, 1993) is defined as the

temporal correlations between spatially remote neurophysiological events, while

effective connectivity (K. Friston, Frith, & Frackowiak, 1993) is the influence one

neuronal system exerts over another. Functional connectivity is an operational

definition while effective connectivity is not. In other words functional connectivity

is simply a statement about the observed correlations and does not provide any

insight into how these correlations mediate. The distinction between the two

connectivity measures is equivalent to the distinction between correlations and

causality. The exact definition of effective connectivity is vague as it is always tied

to some model of the influence one neuronal system exerts on another. Therefore

the validity of effective connectivity is tied to the validity of the model used

to describe it. Different models can be used to describe effective connectivity

including linear, non-linear models or temporal precedence models like Granger

causality models (Roebroeck et al., 2005). Effective connectivity analysis is more

difficult to study compared to functional connectivity analysis and usually follows

it.

6.2.7 Activations in the absence of task - resting statenetworks

Functional networks in the presence of task is intuitive to understand and can

be easily found using GLM and statistical analysis discussed earlier. What is

surprising and counter-intuitive is the presence of such functional networks in

the absence of any task (Fox et al., 2005). Furthermore these networks are

reproducible across datasets and subjects suggesting a common architecture (Biswal

et al., 2010). The brain is constantly active with high levels of activity even when

one is not engaged in a focused mental activity. These functional networks also

referred to as resting state networks (RSNs) are large amplitude spontaneous

low frequency (<0.1 Hz ) fluctuations in the fMRI signal that are temporally

correlated across functionally related areas. There are 8 to 10 functional networks

(Figure 6.3) which show high test-retest reliability (Zuo et al., 2010) and replicability

(Shehzad et al., 2009). While there is a common architecture shared by these

RSNs, each individual’s functional network exhibits unique features and it can

provide quantitative phenotypes for molecular genetic studies and biomarkers of

developmental and pathological process in the brain. Therefore, the collective set

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of these functional networks is referred to as functional connectome in line with

human genome by Biswal et al. (2010). RSNs have already been quite useful in

characterizing pathology and behavior (Zhou et al., 2010; Harrison et al., 2008; Yu-

Feng et al., 2007), but their role in SD vulnerability has not been explored. Given

our prior findings using the DDM, we believe the intrinsic functional differences

will be revealed using the RSN analysis. In the next section we will discuss various

ways to find these RSNs.

Figure 6.3: Eight of the most common and consistent RSNs identified by ICA. (A)RSN located in primary visual cortex; (B) extrastriate visual cortex; (C) auditoryand other sensory association cortices; (D) the somatomotor cortex; (E) thedefault mode network (DMN), deactivated during demanding cognitive tasks andinvolved in episodic memory processes and self-referential mental representations;(F) a network implicated in executive control and salience processing; and(G,H) two right- and left-lateralised fronto-parietal RSNs spatially similar tothe bilateral dorsal attention network and implicated in working memory andcognitive attentional processes. Courtesy of Beckmann et al. (2005).

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Figure 6.4: spatial-Independent Component Analysis (sICA) decomposition ofgroup data.

6.2.8 Finding Group and Subject RSNs

Finding RSNs is not as straight forward as finding task related functional connectivity

since there is no task to correlate with. There are two distinct approaches to

handle the problem:

(i) seed based analysis, and

(ii) multivariate analysis.

Each has its own advantages and limitations. In case of seed based analysis a seed

voxel or a cluster of voxels is chosen and its (average) time-course is correlated

with rest of the brain. While simple to compute, seed based correlation analysis

has serious limitations. The obvious one is the selection of seed voxels unless

there is a strong prior hypothesis based on which a selection of seed voxels can be

done there is no objective way of specifying them. Other not so apparent problem

is effect of nuisance variables like global brain signal (Zarahn et al., 1997), motion

and physiological noise which while of no importance to the analysis can have

strong distributed influences on large parts of the brain, and therefore can severely

contaminate the seed based analysis. The only way to alleviate this problem is

to regress out the time-course of nuisance variables from the fMRI data. Correct

estimation of these nuisance parameters is essential for the success of seed based

correlations.

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Multivariate methods act on the whole data matrix decomposing it into components.

While there are many matrix decomposition methods the one that makes most

sense in this context is spatial Independent Component Analysis (sICA) (van de

Ven et al., 2004). In sICA the data matrix is assumed to be a linear mixture of

spatially independent maps and their corresponding time-courses. In other words

if X is the data matrix in Rm×n where m is the number of time points and n is

the number of voxels, then

X = AS (6.1)

Where S contains statistically independent spatial maps in its rows and A

which is the mixing matrix contains the associated time-course in its column.

Sources are estimated by iteratively optimizing an un-mixing matrix W = A−1

so that some measure of independence is maximized between the rows of S =

WX. Kurtosis and negentropy are commonly used measure of independence.

The standard ICA model presents problem in thresholding as it lacks any noise

model. Consequently statistical significance of estimated sources is no longer

possible within the framework of null-hypothesis testing. A probabilistic ICA

(pICA) model has been developed (Beckmann & Smith, 2004) that includes a noise

term in the standard ICA model. By providing measures of error for estimated

components statistical inferencing and thresholding becomes possible. Beckmann

& Smith (2004) used a Gaussian-Gamma mixture to model the noise and signal

and then use an alternative hypothesis test to threshold. Usually single subject

data is not sufficient to get good estimates of S, therefore data from multiple

subjects are temporally concatenated (Figure 6.4) to get good estimates. Subject

ICA maps can then be obtained using dual regression (Beckmann et al., 2009).

Working of dual regression is simple; the ICA group maps are used as regressors

to estimate individual time-course. These estimated time-courses are then used

as regressors to estimate individual component maps. This is shown graphically

in Figure 6.5. Now that we have a good background of functional connectivity

and RSN analysis, in the next section we will move onto the experiment.

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Figure 6.5: In dual regression individual time-courses are estimated first, whichare then used to estimate individual spatial maps. Image courtesy of Beckmannet al. (2009)

6.3 Materials and methods

6.3.1 Participants

Twenty one right-handed, healthy adults (12 females, mean age = 21.3 years,

stdev = 1.3 years) participated in this study. Informed consent was obtained

from all participants in compliance with a protocol approved by the National

University of Singapore Institutional Review Board. These participants were part

of a larger study investigating the impact of SD on selective attention (Chee &

Tan, 2010). They were selected from a pool of university students who responded

to a web-based questionnaire and met the following criteria:

• age from 18 to 35 years,

• be right-handed,

• weigh between 45 to 95 kg,

• have good habitual sleep (6.5 to 9 hours of sleep every day),

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• have no history of sleep or psychiatric or neurological disorders and have no

history of severe medical illnesses.

The sleeping habits of all participants were monitored throughout the 2-week

duration of the study and only those whose actigraphy data (Actiwatch, Philips

Respironics, MA, USA) indicated habitual good sleep (i.e., they usually slept

no later than 1:00 AM and woke up no later than 9:00 AM) were recruited

for the study. All participants indicated that they did not smoke, consume any

medications, stimulants, caffeine or alcohol for at least 24 h prior to scanning.

6.3.2 Study procedure

Participants made three visits to the laboratory. The first was a briefing session,

during which they were informed about study protocols and requirements. Each

participant then underwent two experimental sessions, once during rested wakefulness

(RW), and once following a night of SD. The order of sessions was counterbalanced

across participants, and sessions were separated by 6 to 9 days in order to minimize

the residual effects of SD in those who completed the SD session first.

For the RW session, participants arrived at the laboratory at 07:30. One

PVT assessment was then administered at 08:00. For the SD session, participants

arrived at the laboratory at 19:30, having stayed awake the entire day without

napping. They were subsequently monitored in the laboratory until 05:00 the

next morning. The PVT was also administered every hour from 20:00 till 5:00

next morning (10 sessions). Participants also rated their subjective sleepiness on

the 9-point Karolinska Sleepiness Scale and mood on a 10-point Likert-type scale

after each PVT assessment. Between each test bout, participants were permitted

to engage in non-strenuous activities such as reading and watching videos. Similar

to previous chapters, participants were then split into two groups-vulnerable and

resistant, based on a median split of the total number of lapses in the last sessions

of SD.

6.3.3 Imaging methods

R-fMRI scan images were acquired on a 3 Tesla Tim Trio system (Siemens,

Erlangen, Germany) fitted with a 12-channel head coil. Foam padding was used

to restrict head motion. A gradient echo-planar imaging sequence was used with

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a TR of 1.5 seconds, TE of 30 ms, field of view of 192 × 192 mm and matrix

size of 64 × 64. Parallel image reconstruction with GRAPPA was enabled and

an acceleration factor of 2x was engaged. Twenty eight oblique axial slices (4mm

thick with 0.4 mm inter-slice gap) parallel to AC-PC line covering whole brain

were acquired. High resolution coplanar T1 anatomical images were also obtained

for co-registration purposes.

6.3.4 Data analysis

The acquired R-fMRI images were preprocessed using FSL version 5 (S. M. Smith

et al., 2004). The following preprocessing steps were applied: motion correction

(Jenkinson et al., 2002), removal of non-brain structures from echo planar imaging

volumes (S. M. Smith, 2002), spatial smoothing using Gaussian kernel of 4mm full

width half maximum (FWHM) to increase signal to noise ratio, high pass temporal

filtering of FWHM 150s to remove low frequency drifts and low pass temporal

filtering of FWHM 5s to remove any high frequency noise. The functional scans

were then registered to MNI152 standard space by using affine registration at

resolution of 4mm (Jenkinson et al., 2002).

For each subject, the preprocessed and normalized fMRI images were concatenated

across time to form a single 4D image. This group data was then processed using

pICA (Beckmann & Smith, 2004). In our analysis we restricted the ICA output

to 20 components, which is consistent with previous work on low dimensional ICA

(S. M. Smith et al., 2009).

The default mode RSN and two left and right frontoparietal RSNs were identified

by visual inspection. Subject-specific RSNs were obtained from group RSNs using

dual regression (Beckmann et al., 2009). To obtain group statistical maps, voxel-

wise random effects analysis was performed using a one sample t-test. Regions

of interest (ROIs) were determined using joint expected probability distribution

(Poline et al., 1997) with height (p<0.001) and extent (p<0.001) thresholds,

corrected at the whole brain level. Participants were then split into two groups

vulnerable and resistant, based on a median split of the total number of lapses

in the last sessions of SD. Average connectivity between these two groups was

compared using a two sample t-test restricted (masked) within the network as

defined by the group statistical maps (ROIs). Threshold was set at (p<0.05)

corrected using a threshold free cluster enhancement (S. M. Smith & Nichols,

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2009). The analysis was repeated on R-fMRI scans obtained in the last session of

SD. Finally the difference in connectivity between RW and SD for all participants

was also obtained for those RSNs that showed a statistically significant difference

in connectivity between the two groups.

6.4 Results

Based on the total number of lapses in the last session of SD, the median split was

found at 2 lapses. This resulted in 10 participants in the vulnerable group (lapses

> 2) and 11 in the resistant group (lapses ≤ 2). The number of lapses in RW

was not significantly different between the two groups (t19 = 1.75, p=0.10, n.s.).

Group ICA statistical maps corresponding to DMN and frontoparietal networks

for both groups are shown in Figure 6.6.

6.4.1 Functional connectivity differences between vulnerableand resistant groups in RW

Resistant participants had significantly stronger (p<0.05, corrected) connectivity

compared to vulnerable participants in the right-frontoparietal RSN (see Figure 6.7)

when they were well-rested. The cluster showing significant differences in connectivity

was located in Brodmanns area (BA) 40 in the ventral right posterior parietal

cortex (rPPC) (peak location MNI: 40,-52,40; cluster size 512 mm3). There were

no significant differences in RSN connectivity within the DMN network.

6.4.2 Functional connectivity differences between vulnerableand resistant groups in SD

Resistant participants continued to have stronger connectivity (p<0.05, corrected)

in the right-frontoparietal RSN (see Figure 6.8) compared to vulnerable participants

after SD. The cluster of significant difference in connectivity was also located in

the ventral rPPC (BA 40), the same region identified in the RW state. However,

the cluster size was bigger after SD compared to RW (1472 mm3) and the peak

location was slightly shifted (MNI: 42,-54, 32). No other significant clusters were

found.

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Figure 6.6: Group ICA statistical maps corresponding to the default mode networkand left and right frontoparietal networks are shown. The statistical maps werethresholded using joint expected probability distribution with height (p<0.001)and extent (p<0.001), corrected at the whole brain level.

6.4.3 Functional connectivity changes from RW to SD invulnerable and resistant groups

Only the right-frontoparietal RSN showed a difference in connectivity between the

vulnerable and resistant groups both in RW and after SD. Figure 6.9 highlights

regions of the brain that displayed significant (p<0.05, corrected) decline in connectivity

across states (from RW to SD) for the right-frontoparietal RSN across all participants.

Large parts of the orbitofrontal cortex (OFC) and parts of the left PPC also

showed significant (p<0.05, corrected) decline in connectivity after SD (Table 6.1).

The rPPC region however did not show any significant decline in connectivity with

SD. There were also no significant differences in connectivity decline between the

99


Figure 6.7: Locations within the right-frontoparietal RSN (demarcated by redboundary) which show significantly stronger connectivity for resistant subjectscompared to vulnerable subjects in baseline rested wakefulness condition. Errorbars show Mean ± SEM.

resistant and vulnerable groups.

Table 6.1: Clusters showing significant decline in connectivity in the right-frontoparietal RSN across states (RW SD) for all subjects.

Anatomical RegionPeak Voxel

Volume (cc)BA T x y z

Middle frontal gyrus right 10 5.6 38 54 -8 22.6Middle frontal gyrus left 10 6.2 -26 66 -12 8.5Ventral left-PPC 40 5.8 -34 -62 36 3.3

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Figure 6.8: Locations within the right-frontoparietal RSN (demarcated by redboundary) which show significantly stronger connectivity for resistant subjectscompared to vulnerable subjects after sleep deprivation. Error bars show Mean± SEM.

6.5 Discussion

To our knowledge, this is the first study that has examined differences in baseline

functional connectivity between individuals vulnerable and resistant to sleep loss.

We found statistically significant differences in functional network connectivity

between participants vulnerable and resistant to SD, even in the well-rested condition.

The vulnerable group had significantly lower connectivity compared to the resistant

group in the ventral rPPC area (BA 40) in the right frontoparietal RSN. Interestingly,

this difference in connectivity persisted after SD and did not show any further

decline.

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Figure 6.9: Locations within the right-frontoparietal RSN that show significantdecline in connectivity across the rested wakefullness and sleep deprived states(RWSD), for all subjects.

6.5.1 Reduced functional connectivity in the ventral rightPPC region as a marker of vulnerability to SD

In the previous chapters we have shown the importance of mean drift diffusion

parameter (which models the rate of accumulation of noisy evidence in a one choice

decision task like the PVT) in discriminating between vulnerable and resistant

subjects. The posterior parietal cortex (PPC) has been shown to play a vital role

in such decision making processes (Platt & Glimcher, 1999; Snyder et al., 1997).

In addition, the ventral PPC region is also implicated in visuospatial attention

(Babiloni et al., 2007). Specifically it enables attentional capture and reorientation

to salient and behaviorally relevant stimulus and is known to have a dominant right

hemispheric lateralization (Hodsoll et al., 2009; Hutchinson et al., 2009; Mevorach

et al., 2006; Stevens et al., 2005). The observed difference in connectivity of the

ventral rPPC within the right frontoparietal RSN, which remained unaffected by

SD could be a marker of an intrinsic property of the group which exists irrespective

and independent of state (RW or SD). This complements our findings in the

previous chapters that DDM parameters in the rested conditions are predictive of

vulnerability to SD.

6.5.2 Role of frontoparietal network

Connectivity differences were primarily observed in the frontoparietal network.

Activations in the frontoparietal network regions are implicated top-down biasing

of visual attention. The frontoparietal network is also known to flexibly couple

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with the DMN and its anti-correlated dorsal attention network (DAN) typically

activated during a task (Spreng et al., 2010). The frontoparietal network regions

have also been implicated in differential performance between vulnerable and

resistant participants. Chee & Tan (2010) observed increased signal in frontoparietal

regions on a visual selective attention task with varying stimulus contrast to

modulate perceptual difficulty. It was further observed that sleep-deprived vulnerable

participants showed reduced top-down frontoparietal signals with this reduction

being particularly significant during SD lapses. Here, we also observed a reduction

in connectivity in large portions of the frontoparietal RSN after SD which supports

previous findings. This reduction was however not significantly different between

vulnerable and resistant participants.

6.6 Conclusions

This study sought to examine functional connectivity differences between individuals

vulnerable and resistant to sleep loss prior to SD exposure, without the need

for participants to perform a particular task. We demonstrated that there were

significant differences in resting-state connectivity within the right-frontoparietal

RSN between vulnerable and resistant groups. Furthermore, this difference in

connectivity did not appear to be influenced by SD. The brain regions showing

significant difference between the groups are implicated in visual attention and

visuo-motor integration, regions crucial to information accumulation and visio-

motor integration. It is clear that the differences between vulnerable and resistant

group are intrinsic properties of the brain as revealed by both this functional

connectivity analysis as well as previous DDM analysis. These findings will help

elucidate a better understanding of subject-specific vulnerability to SD.

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Chapter 7

Summary and Outlook

7.1 Summary

Sleep is an important aspect of human life with nearly 30% of life spent sleeping. It

has strong bi-directional relationship with health. Sleep or rather the lack of it, has

enormous economic and health impact. The modern lifestyle is only exacerbating

the situation which is evident from the progressively decreasing sleep duration.

Given that cognitive effects of sleep deprivation vary considerably but stably across

subjects, it is of paramount importance that the mechanisms that underly this

differential vulnerability be understood. While some work has tried to understand

the mechanisms of differential vulnerability, progress has been slow, and due to

the relatively recent discovery of the phenomenon (2004) there is huge scope for

further research. Guided by the DDM, this thesis provides a fresh perspective to

the problem.

We start off by first making improvements in the estimation and simulation of

the DDM and understanding its limitations in Chapter 3. The current implementation

of the model is considerably simpler and faster than previous methods. We

believe, with the current improvements and ease of implementation, the clinical

community will find the model attractive for application to wide range of experiments.

In chapter 4, we explore if the DDM parameters are discriminative between the

vulnerable and resistant groups. We observed that not only were the DDM

parameters significantly different between the vulnerable and resistant, they were

capable of measuring these differences even when the standard measurements like

mean RT and lapses failed to measure any such difference. We showed using

simulations that decision and non-decision components of the DDM can trade

off with each other, resulting in similar overall performance despite significant

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Chapter 7. Summary and Outlook

differences in the underlying decision components. This tradeoff is especially

important at high drift rates which is observed in the baseline conditions before

sleep deprivation. We also observed that subjects with lower diffusion drift were

also affected more by sleep deprivation, hinting at possible lower availability of

cognitive reserve for information processing. We further showed that a simple

logistic regression based classifier could classify the upper and lower tertile of

subjects ranked by their vulnerability to SD using the DDM parameters. We

systematically carried this analysis further in Chapter 5. We used a set of features

including standard summary statistics, DDM parameters and wavelet parameters.

A minimum redundancy maximal relevance measure in conjunction with wrapper

based heuristics was used to find a set of most discriminative features. DDM

parameters, ∆RT > 250 and wavelet features were selected as most important

features. Using the best set of features we achieved an accuracy of 77%. This

was achieved using rested vigilance data collected prior to sleep deprivation in

the late evening, corresponding to the wake maintenance zone when the circadian

drive to remain awake is near its peak. The model trained on this dataset and

then tested on a completely independent dataset taken under completely different

experimental conditions showed a respectable accuracy of 71%. When the feature

selection process was applied independently to this dataset, again the same set of

features appeared as most important. An accuracy of 82% was achieved on this

dataset. We also tested the repeatability of the classification by testing the model

on a set of follow up subjects. Our results showed that prediction of cognitive

vulnerability to SD can be reliably and robustly done using DDM parameters

along with ∆RT > 250 parameter.

In Chapter 6, we moved our attention to functional resting state connectivity.

We showed that brain regions involved in visual attention and visuo-motor integration

showed significantly reduced functional connectivity for vulnerable subjects compared

to resistant subjects and these differences persisted after sleep deprivation. These

differences were observed in the resting state in the absence of any task. The

imaging analysis suggests that these differences are intrinsic to the group. The

involvement of the brain regions implicated previously in information accumulation

and visio-motor integration in a task free condition squarely ties the observation

to the results obtained using the DDM analysis.

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7.2 Future Work

In this thesis, we provide a good foundation for future experiments to build upon.

There are two clear directions in which we see future research ahead. One in

the direction of improving vulnerability classification accuracy and other in the

direction of gaining fundamental understanding of the mechanics of differential

vulnerability. Naturally, both directions will complement each other. Before we

discuss these possibilities, there are a few issues which need to be clarified.

7.2.1 Absolute scale for vulnerability

We have shown in Chapter 5 that performance on a PVT is strongly influenced

by the experimental conditions. Defining vulnerability as the relative ranking

within a group which undertook the PVT under similar experimental conditions

is straightforward. For realistic scenarios where vulnerability prediction can be of

use, strictly controlling the experimental conditions could be impractical. Therefore,

a suitable way to compare psychomotor vigilance across groups need to be formulated.

This will in essence provide an absolute scale for quantifying vulnerability. Analytically,

the task of comparing performance of two datasets or subjects is essentially that

of finding the change in threshold or bias. We have shown that the model trained

on one dataset and tested on another dataset which was taken under notably

different experimental conditions worked reasonably well. This suggests that

the information about this bias is embedded within the RT distribution. One

possible way to address the issue is to construct a mapping from DDM parameters

estimated in the sleep deprived condition to an absolute scale of vulnerability. To

be able to achieve this, psychomotor vigilance data for the same group of subjects

in two or more different experimental conditions can be collected. The stability

and reliability of such mapping must be demonstrated fully before it will be given

any credence by the clinical community.

7.2.2 Prediction under field conditions

In our investigation, data was collected under controlled laboratory conditions.

If the experimental conditions are unconstrained, then the sources of variability

will increase which could adversely affect classification reliability and accuracy.

For vulnerability predictions in real world scenarios, the system need to perform

106


well under widely varying experimental conditions. While our results show that

predictions are robust to varying experimental conditions across two datasets it

remains to be seen how it performs on wider sets of data. One of the key difficulties

in vulnerability experiments (or for that matter any experiment involving sleep

deprivation) is the difficulty and cost in acquiring ground truth data, as it involves

subjecting participants to a full night of sleep deprivation. Based on the discussions

with our collaborators, most sleep laboratories measure PVT data after sleep

deprivation as a part of the study. What remains missing is the baseline psychomotor

vigilance data. While relatively less time consuming, collecting rested data from

these subjects is still logistic challenge and financially costly. One way to address

the issue is to build a web/mobile PVT application and reaching out the participants

for whom ground truth is already available to give the PVT at their own convenience

using their own computer or mobile phone. This will naturally provide a realistic

unconstrained data acquisition condition to test the prediction model. We have

made some headway in this regard, which is discussed in Appendix A.

7.2.3 Improving classification accuracy

One of the easiest way to improve classification is possibly by collecting more data

at baseline condition. This can be done either by increasing the duration of each

PVT or by collecting more PVTs. Each route has its own distinct challenges.

The problem with collecting multiple PVTs is that it limits the practicality of

the whole endeavour as sufficient gap must be provided between PVTs to avoid

the effects of time on task. On the other hand, increasing the time of PVTs will

increase the influence of time on task. This is problematic from the point of view

of DDM analysis as the IID assumptions will be strongly violated. Experiments

could be designed to understand how the effects of time on task differ between the

two groups which will shed some light to the problem. Another way to handle the

problem is to explicitly model the effects of time on task within the DDM. Given

the importance ∆RT > 250 metric, an autoregressive model for the drift can be

used: ξt = βξt−1 +error, although this should ideally be dictated by experiments.

In our investigation we used an SVM for classification which is a well established

classification method. As our goal was to show the practicality of classification,

we avoided using more complex classification systems. Small improvements in

classification can be expected by using more advanced classification methods. An

107


analysis comparing a set of state of the art classification systems can be carried

out. Such analysis will be more meaningful as more data is collected.

7.2.4 Understanding the mechanics of differential vulnerability

Our investigation has shown that the information accumulation systems in the

brain play a vital role in the observed performance difference between subjects

after sleep deprivation. It remains to be seen how experimental manipulation

of the functioning of these neural circuits modulates the performance of the

subjects after sleep deprivation. This could be achieved using a transcranial

magnetic stimulation of the neural circuits deemed important and observing how

it affects performance after sleep deprivation. Diffusion MRI can be carried out

to test if the functional connectivity differences observed between vulnerable

and resistant subjects are mediated by underlying structural differences in the

brain. Pharmacological interventions can be carried out to gain understanding

of the molecular basis of differential vulnerability. Clinical experiments involving

humans are costly, time consuming and pose risks to the subjects. We believe

our thesis will provide guidance to the clinical community in designing the right

experiments and asking the right questions.

7.3 Conclusion and Final Thoughts

In conclusion, we streamlined the DDM estimation and simulation process by

deriving a closed form solution for the DDM and studying its statistical properties.

We showed that the resistant subjects had significantly lower diffusion drift compared

to the vulnerable subjects even before any exposure to SD. Such differences were

also observed in the resting state connectivity within the right-frontoparietal RSN

and appeared to be unaffected by SD. The specific brain regions that showed

significant difference between the groups are implicated in visual attention and

visuo-motor integration, regions crucial to information accumulation and visio-

motor integration, suggesting a stable and intrinsic functional and/or structural

basis for the observations. We demonstrated that the various DDM parameters

could act in opposing directions. This is especially important in the rested

condition, when the effect of the individual DDM parameters can completely

cancel each other out resulting in similar observed reaction time responses even

108


when the underlying DDM parameters are significantly different. We systematically

showed that the DDM parameters are the most important behavioural metrics

for predicting vulnerability to SD using rested data collected prior to SD. The

prediction is reliable across time and datasets and robust to varying experimental

conditions.

Research at the forefront of science is increasingly becoming multi-disciplinary

with researchers from widely different fields collaborating together to solve a

problem. The advantage of such diverse collaborations is that the researchers bring

very different perspectives of looking at the same problem and consequentially

very different solutions to address them. We believe our work will attract a wide

range of researchers to study vulnerability to sleep deprivation. Predicting and

understanding the dynamics of differential vulnerability to sleep deprivation is

only going to be progressively more important in the increasingly sleep deprived

society.

109

Appendix A

Effective PVT Data Acquisition

In this work we have shown that subjects vulnerable to sleep are fundamentally

different from resistant subjects as revealed by DDM and imaging studies. Furthermore,

the vulnerability can be reliably predicted under varying experimental conditions

using baseline PVT data. The predictive capability could further improve with

integration of other vital bio-signals like heart rate, body temperature, perspiration

etc. Wearable bio-sensor devices can already measure these signals in conjunction

with smartphones. Therefore, a software based cross platform PVT data acquisition

system could be an invaluable tool to push this research work further to large

population in cost and time effective way. In the next section, we will discuss the

current solutions available and their shortcomings, technical difficulty in developing

such a system and finally we present the solution that we have developed to address

the issue. As the challenges involved are primarily technical, we have moved the

discussion out of the main text of the thesis.

A.1 Existing PVT Data Acquisition Systems

• PVT-192 (Ambulatory Monitoring Inc., Ardsley, New York USA) is the

gold standard in PVT acquisition. It is a hardware based solution with

claimed latency of 1 ms. The currently quoted price of the device is $2500.

• PVT Laptop System (Pulsar informatics inc., Philadelphia, Pennsylvania

USA) for Windows. Currently priced at $1399 per unit.

• Software App (Pulsar informatics inc., Philadelphia, Pennsylvania USA)

available on iPad. Currently priced at $99 per app download plus additional

yearly membership charges to access data on cloud.

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Chapter A. Effective PVT Data Acquisition

• PC-PVT (Khitrov et al., 2014), is a free software based solution available

on Windows platform. The software primarily targets laboratory settings

and is not meant for direct use by the subjects or end user. Being developed

in Matlab, it requires a download of a hefty 400 MB Matlab runtime to

work.

The high price of the commercial solutions make it infeasible to conduct future

experiments on large scale. Additionally, the commercial solutions are available

on limited platforms. The only free solution available currently runs on desktop

system running Windows and is not geared to be used directly by the subjects.

We intend to solve the problem by developing a software solution that is user

friendly and cross platform. In the next section we discuss the primary challenges

in developing such solution.

A.2 Technical Challenges

The key challenge in developing a software based solution capable of running on

general purpose hardware is the requirement of keeping the latencies to within a

few milli-seconds. As an average PVT RT is only 250 ms, and the intra-subject

variability in RT is estimated to be 29 ms (Rupp et al., 2012), for reliable use the

latencies of the system at-least be within 10 ms. In a general purpose hardware

there are three primary sources of latencies:

(i) Latencies due to pooling rate of input device. Standard desktop keyboard

and mouse are pooled at 250 Hz (source: Intel inc.,Santa Clara, California

- USA). High end gaming devices are pooled as fast as 1000 Hz. Smartphones

usually have 100Hz scan rate for touch screens (source: Cypress Semiconductor

Corporation, San Jose, California - USA).

(ii) Latencies due to processing time of the software application. This could

range from less than a millisecond on a desktop system to as high as 100ms

on smartphones.

(iii) Latencies due to refresh rate of display screen. This in almost all cases is 60

Hz.

111


All these latencies can add up to 100 ms or more, which is unacceptable for a

PVT acquisition system. In fact, most smartphones show latencies in this range

for a simple touch-to-respond application (Figure A.1). The problem is further

exacerbated by the widely varying hardware and software configuration between

available devices. Careful design of the software is essential to keep the latencies

in check.

Figure A.1: Latencies in response for some of the flagship tablet devices. Courtesyof Agawi TouchMarks, used with permission.

A.3 Design Considerations and Possible solutions

As the PVT involves a relatively simple animation, the processing latencies can be

kept minimal by utilizing hardware acceleration for graphics using multi platform

Open GL libraries (Silicon Graphics, California - USA). Latencies due to pooling

rate of the input device and refresh rate of the display device are strongly tied

to the hardware. While these cannot be reduced at a software level, they can

be accounted for. For this, precise timing information is required for peripheral

devices.

Fortunately, for display devices there is a vertical blanking interval, also known

as VBLANK, which is the time difference between two consecutive screen refreshes.

112


Open GL exposes the VBLANK via glFinish() API (Application Programming

Interface) call. By precisely measuring the delay in the animation, reaction time

measurements will account for display latencies. Similarly, if the average latency

for input device is known, it can be subtracted from RT to reduce average latency.

Unfortunately, neither timing information nor pooling rates are available for input

devices on most platforms. As a result, they will continue to affect the latencies

of the system. One possible way to solve the problem is to hardcode the latencies

on a per system basis. This could be possible on certain devices (for instance on

Apple, due to limited hardware models).

Other than keeping latencies to minimum, another important design consideration

is the choice of development environment. The space of computing devices is

highly fragmented both in terms of hardware and software. Each platform offers

its own choice of software development kit (SDK) and programming language.

Developing independently for each platform is prohibitively costly and time consuming.

Moreover, it constitutes an essential wastage of resources as the same task has

to be repeated in multiple programming languages. After careful considerations,

C++ along with the Qt user interface framework (Digia, Helsinki - Finland) was

selected for developing the system. C++ allows high performance and both high

and low level access to underlying hardware and Qt framework makes it easy to

develop the system in a platform agnostic way.

A.4 Quick PVT: A Cross Platform PVT Software

Solution

The PVT software solution named Quick PVT was designed targeting Desktop

systems running Windows, Linux or iOS and also mobile platforms including

Android, iOS, Windows Phone and Blackberry. Screenshot of the application

running on a Windows system and on an Android phone is shown in Figure A.2.

The application size varied between 14 to 20 MBs. Special attention was paid

to keep the user interface minimal and very easy to use for the subjects and end

users. As the data is generated at the user end, to avoid any data tampering,

the SHA1 hash of the original data along with a secret key is stored. This would

allow the researchers to validate the authenticity of the data received.

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A.5 Conclusion

We believe our work will be of considerable interest to both the clinical and

machine learning community. Looking at the present trends, it appears that

fatigue risk management using distributed and cloud enabled computing devices

will be an emerging industry. Consequently, the Quick PVT software will help

researchers to take our work further in this direction. We have left the precise

measurement of observed latencies and other fine-tuning of Quick PVT on different

platforms as a future exercise.

Figure A.2: Screenshot of Quick PVT running on a Desktop system on Windowsand on a smartphone running Android.

114

Publications

(i) Patanaik, A. and Zagorodnov, V. “Connectivity between visual resting state

networks predicts vulnerability to sleep deprivation.”, Organization of Human

Brain Mapping, 2012

(ii) Patanaik, A., Zagorodnov, V. and Kwoh, C. K. “Parameter estimation and

simulation for one-choice Ratcliff diffusion model.”, Symposium on Applied

Computing-Association for Computing Machinery (SAC-ACM) Special Interest

Group on Applied Computing (SIGAPP), 24-28 March 2014. doi: 10.1145/2554850.2554872

(AR: 24%, Tier: A)

(iii) Patanaik, A., Zagorodnov, V., Kwoh, C. K. and Chee, M. W. L. “Predicting

vulnerability to sleep deprivation using diffusion model parameters.”, Journal

of Sleep Research, 2014. doi: 10.1111/jsr.12166 (IF: 3.8)

(iv) Patanaik, A., Kwoh, C. K., Chua, E. C. P., Gooley, J. J. and Chee, M. W.

L. “Classifying vulnerability to sleep deprivation using baseline measures of

psychomotor vigilance”, Sleep, 2015; 38(5) (IF: 5.1)

115

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